# Guitar/Guitar Maintenance and Storage

Properly maintaining a valuable guitar keeps it valuable. Guitars can
take a lot of abuse, especially if you play live shows and tour, and
even if it pretty much \"sounds the same,\" lack of maintenance may
suddenly render the instrument unplayable. You don\'t need to carefully
examine a guitar every day, but occasional check ups keep it
good-sounding and ready to play.

# Storage

The easiest way to take care of your guitar is to store it properly. The
more expensive the guitar, the better your storage should be. It is
generally accepted that the air humidity should be neither too high nor
too low, thus somewhere in the 45-55% range, and the temperature of the
area should be about 65-75 °F (18-24 °C). These two factors are the
biggest threat to an instrument, because changes in moisture and
temperature can cause permanent warping of the neck and other critical
parts. For guitars made out of solid wood, it is advised to use a
humidifier to prevent cracks and damage from weather change.

On the other hand, guitars made out of multi-layered(plywood) wood,
typically in budget guitars, can withstand relatively more humidity and
temperature changes. Keeping the guitar in a case away from direct
sunlight can help with increasing the life of the guitar.

## Storing Environment

The surest way to keep your guitar in good shape is to remember this
simple rule: *Do not expose the guitar to any climate condition that you
would not want to be exposed to.* If you keep this in mind, your
instrument will likely last years and years.

Avoid large or rapid changes in humidity. Like your body, the guitar
gets used to the climate it is in, and suddenly changing it causes
stress. Humidity is the most dangerous thing that attacks an instrument,
because when wood gets wet, the cell walls become softer and it is more
easily bent. Often, the strings themselves are enough to bend the neck.
Also, if the humidity stays way too low, then the wood will crack and
the structure will weaken.

Temperature on its own is less damaging to the guitar. Wood is generally
tolerant to changes in temperature, and for the most part it expands and
contracts together. Extreme temperatures or rapid temperature changes,
however, can cause serious damage, especially when combined with extreme
humidity. Changes in temperature also affect strings, especially nylon
strings. A significant change in temperature typically detunes the
strings. Other areas to watch for temperature related damage are any
glued joins, like where the neck meets the body, or the fretboard
attaches to the neck.

Never expose your instrument to extreme temperatures for a long time.
For instance, leaving your guitar in a car in the summer all day, or
leaving it outside overnight are sure ways to completely destroy your
instrument. Also keep the guitar out of direct sunlight as much as
possible, because it makes the wood more brittle and can change the
color of the instrument.

Keeping your instrument strung and in tune is another good way to make
sure that you don\'t harm your instrument. Strings put tension on the
neck. Excessive tension may make the neck bow if the guitar isn\'t kept
in tune.

## Methods of Storage

First, a simple tip: If one is playing the guitar and wishes to put it
briefly aside to look for songs or notes or the like, then the best
repository is the couch, the bed, or the floor (with carpet or with the
guitar bag as a cushion). The basic rule reads: What lies down, cannot
fall down. A guitar gets most of its scrapes because one leans it
against a wall, or against a table edge, and then it is knocked over
from the slightest contact or draft of air. If one had put it down, this
would not have happened.

### Wall Hooks

These are most often seen on the walls of guitar stores that must
display dozens of instruments in a small amount of wall space. These are
good, because they keep the guitar out of the way but openly displayed.
These are just a U-shaped piece of metal, covered in rubber or soft
plastic.

The piece screws to the wall, and the headstock rests snugly between the
two pieces of metal. In regards to plasterboard walls be sure to drill
into the timber studs. The weight of the guitar poses no risk to the
neck or headstock. When you select a spot to hang a guitar, avoid
outside walls. These are subject to more temperature change, which can
risk damage to the instrument.

### Guitar Stands

For the most part, guitar stands look similar to a wall hook, except
instead of all the weight being on the headstock, most is on the bottom
of the body and the neck is mostly supported to keep the guitar standing
straight.

Each type of guitar has a specialized type of stand. For example, an
Ovation guitar, which has a rounded plastic back, requires a differently
shaped stand than a Fender Stratocaster or a regular acoustic guitar.
Regardless of what type of stand you get, you should always make sure
that it holds your guitar firmly. Some stands also have a locking
device, which adds an extra level of security.

One problem you might encounter (although it is rare) is that the
lacquer used on your guitar has a reaction with the rubber used to coat
the stand. When you buy a new stand, you should examine the guitar every
few days and look for discolorations or weak spots. As is often the
case, serious guitar damage is easiest to stop before it starts.

### Cases

!A Les Paul style guitar in a hard
case{width="150"}

There are two main kinds of guitar case, gig bags and hard cases. Gig
bags are a favorable kind of keeping, because they give a good amount of
protection, and they are also light to carry. Some often have backpack
style shoulder straps. Gig bags do not protect against temperature
changes very well. Hard cases, in contrast, provide excellent protection
against temperature, humidity and physical damage. Hard cases are also
essential for taking a guitar on an air plane, or for long journeys.

Compared to other methods of storing, cases are by far the most secure,
and this is especially true of hard cases. If the guitar is secured
properly in the case (almost) nothing can happen to it under standard
conditions.

The biggest (and perhaps only) disadvantage of a case is that you cannot
openly display your instrument the way you can on a wall hook or stand.
Price is also a disadvantage, because although gig bags can be bought
relatively cheaply, hard cases are expensive. Still, a cheap bag for an
expensive guitar is a poor investment.

When you buy a case, you absolutely *have to* make sure that the guitar
fits in the case. Gig bags are a little more forgiving, but you will not
get a guitar to fit properly in a hard case that is too small. When you
pick up the case, give it a little bit of a shake, and you should not
feel or hear the guitar moving around very much.

### Cabinets

Storing a guitar in a glass-front cabinet shows it off, and can help
keep it at the proper humidity while being quickly accessible. The cost
and required floor space may not be practical for most guitarists. One
alternative is Musik Tent™, an Instrument Humidor that is portable and
lightweight that uniformly humidifies, stores and displays guitars in
wet and dry environments.

# Maintenance

## Body

The body often takes the most abuse, simply because it is the biggest
target. To help keep a fine instrument in good condition, wipe it down
with a soft cloth after playing. Don\'t use water-based furniture
sprays. You can buy specially treated cloths and sprays for guitars at
almost any music store. Dirt, sweat, and often small nicks and scrapes
can just be cleaned up with a cloth, little bit of warm water. Murphy\'s
Oil Soap can be used to clean the whole guitar. Wipe the strings with a
clean cloth. Some guitarists advocate wiping the strings with warm
water, but be sure to protect the fingerboard from moisture. Consider
wiping your guitar strings off every time you play. Oils and dirt left
on your strings make them asymmetrical---as opposed to evenly
cylindrical vibrating bodies. They may even wear out a bit faster. A
soft, natural fiber or \"microfiber\" cloth works the best and is safest
for the finish. Clean the frets as you do the rest of the guitar. If
necessary, carefully use 0.001 steel wool to get grime off the
fingerboard next to the frets. You can also gently go over the frets to
take off any minor nicks.

If you have a stained or lacquered body, you can treat it with a little
bit of furniture polish. However, if you have a guitar with an untreated
body, be extremely careful with polish. For these types, it is better to
find some sort of cleaning oil or wax, since they help prevent hair-line
cracks from developing. After cleaning, the body must be absolutely dry,
because if the wood gets over-moistened, the tone of the guitar will
begin to degrade.

## Neck and Fretboard

The neck is probably the most important part of the guitar, especially
if you want to play it for a long time. Unless the guitar is stored for
extended periods of time, the tension of the strings will always be
pulling against the neck and stretching it away from the body. If stored
for a long period of time, strings should be loosened, to reduce the
tension on the neck. If the guitar gets moist, this neck warping happens
even faster. Sometimes warping can be fixed by adjusting the truss rod,
but this only prolongs the death of the instrument, and can\'t really
fix the problem.

You can also oil or wax the fretboard, but you should first determine
whether the fretboard is stained or painted, and use the appropriate
protection. Always remember that using too much cleaner is always worse
than using no cleaner at all, and always rub it in slowly.

Another drastic way to repair a warped neck on acoustic guitars is take
all the strings off, and place a small glass of water into the body.
Then, keep the body in place and put a small amount of weight (1 or 2
pounds) on the neck and let it bend back into the proper shape. When it
has been corrected, remove the water, keep the weight on and let the
guitar dry. Hopefully the neck will remain in the correct position,
however it will be much more prone to warping from that point on. Since
this procedure is somewhat accident prone, some manufacturers offer
special instrument air moisturizers, which you can put in a case, or on
a specific area of the neck. These generally allow for a higher rate of
success.

The fretboard is usually made from untreated wood, and it should be
cleaned regularly, before dirt begins to build up. Usually a good time
to do this is when you change your strings, which should be every month
or two. You need to clean the wood between the frets, and the simplest
way is using a clean damp cloth or some very fine steel wool. Use some
water with a little bit of detergent to make cleaning easier. If you use
steel wool, you can also clean up the edges of worn frets, which is
important because smooth frets improve the life of your strings. If the
frets are really worn down, they can be replaced, but this is generally
not a good project to undertake yourself.

It is very important that the neck is not wet after cleaning, because
water damages the structure of the neck. Your cleaning cloth should be
damp, not soaking wet. After cleaning, you can also apply a coat of
furniture polish to seal the wood.

## Strings

A set of strings wear down slowly, if you maintain them properly. Since
regular playing does some amount of damage to the strings, it\'s a good
idea to change regularly. Full sets of strings should be replaced at the
same time. If you only replace one string, the others are likely to
break soon, the strings will have different tones, and the opened pack
of strings will begin to corrode. When changing a set of strings, some
guitarists recommend replacing them one at a time, rather than removing
all strings at once, to maintain tension on the neck (i.e., remove and
replace the first string, then the second, then the third, etc.).

If you do not clean your strings, they may become dull, and even begin
to rust. Dirty strings also damage the frets themselves, because the
grime and rust makes the strings more coarse. There are many types of
string cleaners. Pharmacies sell boxes of individually wrapped
alcohol-saturated pads. These are excellent for occasionally
deep-cleaning strings, but protect the fingerboard.

If you clean your strings infrequently, you can just as easily use glass
cleaner to release the sweat from the strings. It\'s easiest to soak a
cloth in window cleaner, and then slip the cloth behind, and clean the
whole length of each string individually. You can tell when a string is
cleaned when you rub the string and the cloth is still clean. Also, you
shouldn\'t let the cleaner remain on the strings, because residue might
damage the string too. As always, protect the fingerboard from moisture.
When finished, wipe the strings with a dry cloth.

## Tuning Mechanisms

Tuning mechanisms are usually chromed, gold-plated, anodized, or
burnished steel. Since steel rusts, especially when it contacts sweat,
these pieces should be cleaned periodically. Properly oiled mechanisms
work smoothly and help keep the strings in tune.

To prevent rust, clean and oil parts regularly. You can use commercial
machine oils from any hardware store, but baby oil or petroleum jelly
works too. Too little lubricant is better than too much---adding more is
easier than removing excess. Two drops is often enough. Avoid making
contact with the wood parts of the guitar because the oil could stain or
discolor. Electrical components (switches, sockets, potentiometers,
etc.) can also go bad if the oil gets in them.

Many tuning machines have a screw in the knob that controls how easily
they turn. The knobs should be snug, with no free play, but they should
not be so tight that they are difficult to turn. Care should be taken
not to over-tighten, as this may break plastic tuner buttons or may
strip the screw thread and require repair or replacement.


# Guitar/Adjusting the Guitar

Many beginning or even intermediate guitarists are unaware that their
guitar should be `<b>`{=html}\"set up\"`</b>`{=html}. The adjustments
described in the adjustment subsections below (along with restringing
and tuning) are called a `<b>`{=html}\"set up\"`</b>`{=html}.

# What difference does a set up make?

When a guitar is set up properly:

-   the guitar will feel and sound its best
-   all the strings will sound with exactly the notes they are supposed
    to
-   all notes will sound correct when played at each fret up and down
    the neck
-   the guitar will be as easy as possible to play
-   strings will break less frequently.

If a guitar plays easily and sounds its best then it\'s easy for the
player to feel successful.

When a guitar is not set up properly:

-   the guitar may not feel or sound quite right
-   some notes may sound correct while some others may sound sharp or
    flat
-   the guitar may be difficult to play
-   strings will break more often
-   damage to the instrument could be incurred unwittingly by the player

## When to Set Up?

When a guitar is brand new and fresh from the factory it may or may not
have had these adjustments done. As a rule, a guitar should be set up
when first purchased (used or new) and again when switching string
gauges. Consider getting a set up anytime the guitar sounds or feels
different than it used to. Perhaps after a guitar travels (altitude
changes, pressure changes, and humidity can affect the wood in the
guitar) and just like changing oil in a car it is a good idea to get a
set up every now and then for maintenance purposes (perhaps twice a
year).

Poor set up may be obvious to a player or it might not. In some cases
the guitar may be unplayable because it hasn\'t been set up. A
maladjusted guitar can cause strange quirks, for instance frets near the
bottom of the neck being too sharp, or can even cause damage (e.g., by
using .012 gauge strings on a nut designed for .009 strings, and the
tension messes up the nut), and it can easily frustrate the player when
their playing is perfectly correct yet things still don\'t sound right.

In particular if your guitar ever becomes difficult for you to play, a
set up will probably help.

It is not absolutely *required* to set up a guitar, but it is
nonetheless a good idea, *especially* if the guitar is to be taken to
the stage. Some people never get their guitar set up. Some get their
guitar set up even when nothing previously seemed wrong with it, then
find such a dramatic change in the guitar\'s playability and sound that
they wish they had set it up sooner.

## How to get a Set Up

These adjustments should generally be done by a professional, qualified
repair person. They require precision instruments, some hard to find
tools, a steady hand, quite a bit of time and know-how.

Virtually all musical instrument stores will be able to perform a
professional set up. Some will do the job better than others. Call a
local music store and ask them \"Do you do set ups for electric (or
acoustic) guitars and how much would you charge?\". Getting a set up
will probably cost from \$30 to \$75 USD.

# Adjustments

## Adjusting action at the bridge

This is a simple adjustment that can usually be performed without
professional assistance. The bridge saddles should be lowered if the
string action is too high, that is, the strings are too far up off the
fretboard. In some cases it may be desirable to raise the saddles for a
higher string action.

Most electric guitars have two small screws on the saddle which can be
used to raise or lower the saddle. Some saddles have screws that can be
rotated using the fingers; others require an allen key. Lower the
saddles too much and the strings might rattle against certain frets
(this may or may not be inconsequential on an electric guitar; listen
through an amplifier). In more extreme cases, pressing a string against
one fret might actually fret the string against a different fret,
usually the one under the intended one. In both cases, filing the frets
might alleviate the problem if the saddle really should be that low.
Otherwise, simply raising the saddle a small amount on the side with the
problem should be fine.

## Filing frets

The frets go with the shape (or contour) of the neck radius (9, 12\",
16\", etc.) Frets can determine how the notes on your guitar sound (i.e.
intonation) and over time and use playing a fretted instrument the frets
will begin to wear out by either changing the shape of the crowns (the
top of the frets, and they are changed by being flattened out or
mis-shaped)or the frets will begin to leave their slots. Fret work
should be done by someone with experience doing this kind of job because
this is a job that can lead to worse problems on your guitar such as
your tonation being worse, action may be higher, strings may buzz out,
and it may require that multiple frets be replaced or further repaired.
Frets come in a variety of sizes as well making them each different to
work on and there are special tools available to do this line of work
but many are expensive and without proper training may not be used
correctly.

## Filing the nut

Filing the nut should only be done by a qualified repair person and is
used to reduce pressure at the nut to allow a heavier gauge of strings
to be used. It may not be necessary if the new strings are detuned lower
(e.g., when switching from .009\'s to .010\'s, the nut will need no
adjustment if the guitar is tuned to Eb-Ab-Db-Gb-Bb-Eb instead of
E-A-D-G-B-E).

## Neck/truss rod adjustment

This particular adjustment has been known to ruin guitars when performed
incorrectly, so here referral to a professional repair person is highly
recommended. A guitar will need a truss rod adjustment if the neck is
not straight. One way to check the straightness of the neck is to play
12th and 19th harmonics on the low and high strings. After sounding each
harmonic, fret the note there and play it again: it should be exactly
the same pitch. If it is not, the neck *may* be in need of adjustment.
However, this may be indicative of an intonation problem as well, which
*can* be fixed without the aid of a repair person; see below. If
adjusting the intonation does nothing for you, give the guitar to a
repair person.

## Adjusting intonation

You may notice each string on the bridge sits in a \"saddle\". Depending
on your setup, you might notice the saddles may be in different
positions: some might be pushed forward and others might be pushed back,
sometimes slightly. The positioning of the saddle effectively changes
the length of the vibrating string. Tune the guitar to concert pitch
with the aid of an electronic tuner, making sure the open strings are
*perfectly* in tune. Play the 9th and 12th fret harmonics, then play the
fretted notes. If the fretted notes are sharp, the string is too short
and the saddle needs to be pushed back toward the base of the bridge. If
the note is flat, the string is too long and the saddle needs to be
pushed up toward the nut. Repeat this procedure for each string.
Adjusting the intonation should be done every few months or at least
twice a year (Every six months interval).


# Guitar/Stringing the Guitar

Aside from the physical shape of the guitar body, strings are the most
important thing for determining the sound of a guitar. New strings sound
bright and full, while old strings tend to sound dull and dead. Many
guitarists believe that strings should be changed regularly, not just
when they break. This is because sweat and dirt corrode the strings, and
over time this degrades their sound quality. Other guitarists believe
that new strings sound much worse than old ones, feeling that a
string\'s tonal quality only improves over time. Individual string
quality may vary drastically from string to string.

When one breaks a string, all of the strings should be changed at once.
This is especially true if the newer string is of a different brand or
gauge. The string\'s manufacturing process, thickness and age all affect
its tone, and one new string being played with a bunch of old strings
can make your guitar sound strange. Players should be advised that
guitars are usually set up for a particular gauge of string. The guitar
will still function fine with a different gauge of strings, however for
optimal sound, the guitar may need to be adjusted. See the chapter on
adjusting the guitar for more
details.

Because there are several different types of guitar, and each type is
designed differently, each type has its own method of stringing. The
type of strings you use mostly depends on what style of music you play
and how long you\'ve been playing. Thinner strings are generally
preferred by beginners, but many experienced players prefer the feel of
thin strings over thicker strings. Please see the guitar
accessories section for details on
different types of strings.

The first thing you always need to do when stringing a guitar is to take
off the old strings. You should *never* just cut the strings of a tuned
guitar in half, because the sudden release of tension on the neck can
damage the guitar. Instead, always turn the tuning pegs to decrease
tension, until the string is so loose that it doesn\'t produce a note
when struck, then cut or unwind them. In most cases, the string is bent
at the end where it was inserted, to insure that it would stay during
tuning. Unbend the string, then pull it out of the peg hole. If the peg
end of the string is too bent or curled from the winding, cut the string
on a straight part of the string. This will make it easier to remove the
string from the hole at the other end and reduce the risk of scratching
the body or the bridge while trying to get it out. Slide the string out
of the bridge at the bottom end of the guitar. Some people string one at
a time to make sure the neck sustains tension, or they just take all of
the strings off at the same time.

## Stringing Acoustics

Standard guitars typically have a ball-head peg at the bridge section.
This peg has a hollow shaft, with a groove that allows the string to
come out from the peg.

Typically, the process is as follows:

Unwinding the string

1.  Pick one of the strings, usually either the first or sixth string,
    and begin loosening it. If you have a string winder, put the
    rectangular box over the tuning peg and unwind the string.

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1.  Once the string is loose enough, pull the peg out of the bridge. If
    you have a string winder, it will have a notch that can fit
    underneath the head to pull it out.

Attaching the string

1.  The guitar string should have a ball (or cylinder) end. Put that
    into the bridge hole.
2.  Push the ball-head peg back in. as you do, pull on the string so
    that the peg can hold the string tightly on the bridge end.
3.  Find the hole on the winder, and place the string through it,
    leaving about 5cm out on the other side. For the thicker strings, it
    is recommended to bend it a bit.
4.  Use one hand to hold the string so that the section between this
    hand and the peg is tight. Wind one wound.
5.  Check the tightness again, and try to divert the string so that it
    wind UNDERNEATH the winder hole. Then wind it until you have the
    string encircle the machine head two to three times.
6.  Tune from here.

### Twelve String Acoustic

It has the same principle to the sixth string. But every two strings
were tuned with the same sound, one octave apart. \*(RDT)

### Classical Guitar

To unstring a classical guitar one method is:

1.  Loosen the string by turning the tuning peg
2.  Then at the bridge push the string back into the hole a little, this
    will loosen the \"knot\" enough to unknot and pull the string out of
    the hole.
3.  Then feed the string around the peg loop by loop until the last hoop
    which is inserted through the hole in the peg is available, push the
    string out of the loop, then pull the loop out of the hole.

To string a classical guitar one method is reverse of the unstringing

1.  Bend about an inch of string at one end to form an open loop, push
    that through the peg hole, wrap the other end of the string around
    the peg and through the loop, then pass it down the guitar body to
    the bridge and into the hole there.
2.  Loop back to the neck (about two or three inches) and twist back
    around the string, then you can put two or three twists in which
    should end up on top of the bridge, pull the string from the middle
    of the guitar to draw the twists taut.
3.  Then wind the peg to tighten the string. You should take it easy
    when tuning up for the first time to give the string time to
    \"settle in\", you may also find that the string may go out of tune
    easily for a day or two as it beds in.

## Stringing Electrics

For the 6th string (the low E), take the string out of the package and
insert the end through the bridge of the guitar. Pull it all the way
through until the ball at the end of the string stops it from being
pulled further. This is optional: Make a kink in the string to insure
that it will not slip away from the turning of the peg, (usually about
one or two inches from the peg). Wind the string around halfway and
insert the end through the hole. Pull the string to add tension, so the
string will stay around the peg during tuning. Turn the tuning peg to
increase tension until the string is around the desired pitch, to make
certain it will stay on properly. Check that the string is in the notch
in the nut and the bridge, if it is not, decrease tension on the string
until you can move it into the notch, tune it back up. Do this for the
rest of your strings and you are done!

Another method:

String the low E and other strings as mentioned. Align the tuning peg\'s
hole with the direction of the string and slip it through the peg in the
direction of the headstock. Facing the guitar with the headstock to your
right, pull the string taut with your left hand.

With your opposite thumb and forefinger, twist the string in an \"s\" at
the twelfth fret so that it touches both sides of the twelfth fret. You
will have to let some of the string out to do this. This method tells
you the optimum length of the string to wind around the tuning peg.

Hold the string with your right hand below the tuning peg so that the
pointy end is sticking out the other side. Slowly tighten the peg so
that the string is winding on the INSIDE of the headstock \-- inside
right for E A D, and inside left for G B E. Allow the string to wind
once underneath itself, and then wrap it over top of itself the rest of
the way. Make sure you hold tight as you go so that there is little
slippage later.

If possible, hold the string with your right thumb and middle finger
while regulating the pressure on the string with your right index
finger.

### Tips

-   Note that taking off all strings at once is not recommended if you
    have:

1.  a floating tremolo system (e.g. Floyd Rose II), which can be
    difficult to get the tremolo angle back to the right level when
    restrung;
2.  a bridge which is not fixed (one that will just fall off when the
    strings are removed)

-   Try not to bend the string in the same place excessively otherwise
    the string will break at the bend


# Guitar/Alternate Tunings

!In standard tuning, the C-major chord has three shapes because of the
irregular major-third between the G- and B-strings. Among alternative
tunings, regular tunings have the same shape for chords everywhere on
the
fretboard.

Many guitar players use alternate tunings, which differ from standard
tuning. The use of alternate tunings (non-standard tunings) are found
throughout the history of the classical guitar and are a major factor in
the playing of blues slide guitar. Many alternate tunings involve
downtuning (\"dropping\") strings.

## Dropped tunings

### Dropped D (DADGBE)

The most common alternate tuning is the dropped D (or \"drop D\")
tuning. The lower E string is tuned down to a D. This tuning allows one
to play power chords
on the fourth, fifth and sixth strings with only one finger, and of
course allows for lower bass notes. Used commonly in heavy metal, but
also in nearly every other form of guitar music.

Waylon Jennings used this frequently, even dropping to D with the use of
a scruggs banjo tuner on his famous telecaster in the middle of a song.

### Drop C tuning CGCFAD

This progressive tuning is used in metal but often used by bands who
stick with 6 string guitars to achieve an extra deep tone. Bands such as
CydeDish, Deftones, System of a Down, KSE, August Burns Red and As I Lay
Dying.

### Drop B (BF#BEG#C#)

Alternatively, you can tune to BF#BEG#C#, if you are looking for a deep
heavy hitting sound. This tuning is mostly used by nu-metal bands like
Slipknot, August Burns Red, many deathcore bands, and some death metal
bands.

### Drop A (AEADF#B)

Alternatively, you can tune to AEADF#B, if you are looking for a deep
heavy hitting sound. This tuning is mostly used by nu-metal bands like
Slipknot, August Burns Red, many deathcore bands, and some death metal
bands.

### Double dropped D (DADGBD)

Similar to Dropped D above, for this tuning just drop both \'E\' strings
a full tone. Neil Young often tunes his guitars this way.

## Regular tunings

!The major-thirds tuning G#-C-E-G#-C-E repeats its three open-notes in
the higher octave after three
strings.
!Chords can be shifted diagonally in major-thirds tuning and other
regular
tunings.
!Minor, major, and seventh chords (*C, D, G*): In major-thirds tuning,
major and minor chords can be played with two fingers on two consecutive
frets. The chords have the same shape, unlike the chords of standard
tuning.: In major-thirds tuning, major and minor chords can be played with two fingers on two consecutive frets. The chords have the same shape, unlike the chords of standard tuning.")

Among alternative guitar-tunings, regular tunings have equal
musical-intervals between the paired notes of their successive
open-strings. Regular tunings simplify the learning of the fretboard of
the guitar and of chords by beginning students. Regular tunings also
facilitate improvisation by advanced guitarists.

Guitar tunings assign pitches to the open strings of guitars. Tunings
can be described by the particular pitches that are denoted by notes in
Western music. By convention, the notes are ordered from lowest to
highest. The *standard tuning* defines the string pitches as E, A, D, G,
B, and E. Between the open-strings of the standard tuning are three
perfect-fourths (E-A, A-D, D-G), then the major third G-B, and the
fourth perfect-fourth B-E.

In contrast, regular tunings have constant intervals between their
successive open-strings:

-   4 semitones (major third): Major-thirds tuning,
-   5 semitones (perfect fourth): All-fourths tuning,
-   6 semitones (augmented fourth, tritone, or diminished fifth):
    Augmented-fourths tuning,
-   7 semitones (perfect fifth): All-fifths tuning

For the regular tunings, chords may be moved *diagonally* around the
fretboard, indeed *vertically* for the repetitive regular tunings (minor
thirds, major thirds, and augmented fourths). Regular tunings thus
appeal to new guitarists and also to jazz-guitarists, whose
improvisation is simplified. On the other hand, some conventional chords
are easier to play in standard tuning than in regular tuning.

### Major thirds tuning

!Chords can be translated vertically by three strings, because
major-thirds tuning repeats itself (at a higher octave). Again, the
chords have the same shape, unlike the chords of standard
tuning.. Again, the chords have the same shape, unlike the chords of standard tuning.")
\\Image:First and second inversions of C-major chord on six-string
guitar with major-thirds tuning.png\|right\|thumb\|alt=The C major chord
and its first and second inversions. In the first inversion, the C note
has been raised 3 strings on the same fret. In the second inversion,
both the C note and the E note have been raised 3 strings on the same
fret.\|Chords are inverted by shifting notes three strings *on the same
fret*.

Major-thirds tuning was introduced by jazz-guitarist Ralph Patt in 1964.
All of the intervals between its successive open strings are major
thirds; in contrast, the standard guitar-tuning has one major-third amid
four perfect-fourths.

Major-thirds tuning reduces the extensions of the little and index
fingers (\"hand stretching\"). Major and minor chords are played on two
successive frets, and so require only two fingers; other
chords---seconds, fourths, sevenths, and ninths---are played on three
successive frets. For each regular tuning, chord patterns may be moved
around the fretboard, a property that simplifies beginners\' learning of
chords and that simplifies advanced players\' improvisation. In
contrast, chords cannot be shifted around the fretboard in the standard
tuning E-A-D-G-B-E, which requires four chord-shapes for the major
chords. There are separate chord-forms for chords having their root note
on the third, fourth, fifth, and sixth strings. Major-thirds tuning
repeats its octave after every two strings, which again simplifies the
learning of chords and improvisation; Chord inversion is especially
simple in major-thirds tuning. Chords are inverted simply by raising one
or two notes three strings. The raised notes are played with the same
finger as the original notes.

### All fourths tuning

![In *all-fourths* and standard tuning, the C7 chord has notes on frets
3-8; Covering six frets is difficult, and so C7 is rarely played but
\"alternative voicing\" are substituted instead. In *major-thirds*
tuning, all seventh-chords can be played on three consecutive
frets._tuning_for_six_string_guitar.png "In all-fourths and standard tuning, the C7 chord has notes on frets 3-8; Covering six frets is difficult, and so C7 is rarely played but "alternative voicing" are substituted instead. In major-thirds tuning, all seventh-chords can be played on three consecutive frets.")
!The consecutive notes of all-fourths tuning are spaced apart by five
semi-tones on the chromatic
circle.
!All fourths tuning.

:   E-A-d-g-c\'-f\'

This tuning is like that of the lowest four strings in standard tuning.
Consequently, of all the regular tunings, it is the closest
approximation to standard tuning, and thus it best allows the transfer
of a knowledge of chords from standard tuning to a regular tuning. Jazz
musician Stanley Jordan plays guitar in all-fourths tuning; he has
stated that all-fourths tuning \"simplifies the fingerboard, making it
logical\". Contemporary players are also moving to fourths tuning, noted
players include Alex Hutchings and Tom Quayle.

:   Eb-Ab-Db-Gb-B-E

A less commonly used version of fourths tuning is retaining the regular
tuning on the first and second strings (E and B) and lowering the other
strings by a semitone. This version can make playing classical pieces
more accessible.

### Augmented fourths tuning

### All fifths tuning

#### New Standard Tuning (CGDAEG)

!**New standard tuning**. All perfect fifths (C-G-D-A-E) plus a minor
third
(E-G). plus a minor third (E-G).")
!C major chord in new standard tuning
.
!D major chord in new standard tuning
.
![Open fifths on D in new standard tuning
.

The tuning, invented and introduced by Robert Fripp of
King Crimson, is: C(6th) - G(5th) - D(4th) - A(3rd) - E(2nd) - G(1st).

Basically this tuning is efficient because it utilizes the tuning that
is common is a cello (CGDA) , violin, and mandolin (both GDAE), in which
it is in fifth, from a low C. The second string is a fourth up from the
B to an E, and the first string is a minor third up from the E to a G.

Since the lowest five strings are tuned in fifths, typical fingerings
for chords and scales used on the violin, cello, and mandolin are
applicable here. The minor third between the top strings allow denser
chords in the high range of the scale, and easier access to some
elementary chord tones (typically the thirteenth for chords with the
root note on the sixth string, and the ninth and flat ninth for chords
with the root note on the fifth string, see chord). NST has a greater
range than the Old Standard Tuning, approximately a perfect fifth (a
major third lower and a minor third higher).

Scales across two strings in NST also line up nicely into coherent
tetrachords or four-note patterns that have a visually rational
relationship (whole and half-tone relationships have a remarkable
symmetry that can be easier to learn than the OST whose intervals from 6
to 1 have the (inconsistent) major third thrown in the middle of the
scale.

{{-}}

## Open tunings

### Open A (EAC#EAE)

Alternatively you could tune the guitar to EAC#AC#E

### \"Slide\" Open A (EAEAC#E)

This tuning is identical to Open G tuning but with every string raised
one step, or two frets

### Open C (CGCGCE)

Used notably by Devin Townsend in his bands Strapping Young Lad and
Devin Townsend Project.

### Open D (DADF#AD)

Open D, like all open tunings, produces a major chord (in this case, D
major) when all strings are strummed. Drop the sixth, first, and second
strings down two semitones, and the third string one semitone. It is
also called \"DAD-fad\" after its notes.

Uses the same chord shapes as Open E but is easier on a guitar neck as
the strings are detuned lessening the tension.

Open D is a common tuning for folk, blues, and slide guitar. A variation
of this tuning is open d minor. Open D minor is tuned DADFAD, meaning
the only change is that the F# is tuned down to an F.

#### Chord shapes

Here are some handy chord shapes: G/D: (020120) Em7/D: (022120)

### Open E (EBEG#BE)

Used by Cat Stevens and a popular choice for slide guitarists. Strumming
in the open position yields a Emajor chord. You can easily play any
chord by barring across the neck at different fret positions. This does
however have some disadvantages; mainly that it is slightly more
difficult to play minor chords. Some artists overcome this by tuning to
EBEGBE. This allows both minor and major chords to be played easily.
Because tightening the strings more than is intended can break the
strings or put unneeded stress on the neck, many players opt to tune in
Open D and put a capo on the second fret; the result is the same.

### Open C6 (CACGCE)

This tuning is rarely used. It has been used by Jimmy Page on
\"Bron-Yr-Aur\", \"Friends\", and \"Poor Tom\". J. M. Smig uses this
tuning on \"Fingers In The Forest\".

### Open G (DGDGBD)

This is sometimes referred to as \"Spanish Tuning\", popular with slide
guitarists. Tune the 1st and 6th strings down to D, and the 5th string
to G.

Keith Richards uses this tuning extensively after 1968. (See Brown
Sugar, Honky Tonk Women, Start Me Up) He also removes the 6th string
(low E in standard tuning) because the root of the chord is on the 5th
string in Open G.

## Miscellaneous tunings

### DADGAD

DADGAD (pronounced as a word: \"DAD-gad\"), one of the most versatile
tunings, is named after the tuning of its strings. The sixth, second,
and first strings are dropped two semitones to D, A, and D. Strumming
all the strings open forms a Dsus4 chord; fretting the second fret of
the third string (or muting the third string) produces a D5 chord, or D
power chord. Most songs for DADGAD are in D major, or in G major with a
capo at the fifth fret. Jimmy Page of Led Zeppelin played Kashmir in
this tuning.

### DADADD

This is essentially one huge power chord. Each string neatly divides the
scale in half and it is easy to make simple patterns then repeat them
anywhere on the fretboard.

### Standard E-flat (EbAbDbGbBbeb)

In this tuning, each string is tuned down a half step, or one fret. This
is a popular tuning throughout the history of blues and rock, and many
modern bands perform with it. Stevie Ray Vaughn frequently used this
tuning with heavier strings to get a full, rich tone.

### Standard D (DGCFAD)

Made popular by death metal band Death. Commonly used in metal today.

**Standard C (CFBbEbGC)**

Used by John Petrucci on 3 songs from Dream Theater's Train Of Thought.

fr:Apprendre la
guitare/Accordage


# Guitar/Scale Manual

## Circle Of Fifths

!Circle of
Fifths{width="300"}

The Circle of Fifths is a memory aid for learning all the keys. Here are
some fundamental observations:

Clockwise direction shows all the sharp keys

Anti-clockwise shows all the flat keys

Both directions start at the key of C major which is 12 o'clock

C major is the only key that has no sharps or flats

Clockwise ends with the key of C sharp

Anti-clockwise ends with the key of C flat

C, C flat and C sharp use the same notation letters

Each successive \"hour\" clockwise adds a sharp

Each successive \"hour\" anti-clockwise adds a flat

The name \"circle of fifths\" refers to the interval between the root
note of each successive scale in the clockwise direction.

Anti-clockwise the root note of each scale forms the interval of a
fourth with the root note of the following scale.

## One Octave Major Scales Clockwise

!C major one
octave.png "C major one octave"){width="800"
height="142"} !G major one
octave.png "G major one octave"){width="800"
height="142"} !D major one
octave.png "D major one octave"){width="800"
height="142"} !A major one
octave.png "A major one octave"){width="800"
height="142"} !E major one
octave.png "E major one octave"){width="800"
height="142"} !B major one
octave.png "B major one octave"){width="800"
height="142"} !F sharp major one
octave.png "F sharp major one octave"){width="800"
height="142"} !C sharp major one
octave.png "C sharp major one octave"){width="800"
height="142"}

## One Octave Major Scales Anti-Clockwise

!C major one
octave.png "C major one octave"){width="800"
height="142"} !F major one
octave.png "F major one octave"){width="800"
height="142"} !B flat major one
octave.png "B flat major one octave"){width="800"
height="142"} !E flat major one
octave.png "E flat major one octave"){width="800"
height="142"} !A flat major one
octave.png "A flat major one octave"){width="800"
height="142"} !D flat major one
octave.png "D flat major one octave"){width="800"
height="142"} !G flat major one
octave.png "G flat major one octave"){width="800"
height="142"} !C flat major one
octave.png "C flat major one octave"){width="800"
height="142"}


# Guitar/Chord Reference

# Open Chords

Open chords tend to sound \"richer\" than corresponding barre chords due
to the fact that open strings resonate freely without any hindrance.
Open chords typically sustain longer than their barre chord equivalent.
An open chord is any chord that has one or more strings sounded without
being fretted. Guitarists tend to use this name to refer to the chords
that contain open strings that are played in the first position (first
four frets).

## Open Major chords

```{=html}
<div style="font-family:'Courier New'">
```
File: A major chord for guitar (open).svg\|**A major** File: C major
chord for guitar (open).svg\|**C major** File: D major chord for guitar
(open).svg\|**D major** File: E major chord for guitar (open).svg\|**E
major** File: F major chord for guitar (open).svg\|**F major** File: G
major chord for guitar (open).svg\|**G major**

```{=html}
</div>
```
## Open minor chords

```{=html}
<div style="font-family:'Courier New'">
```
File: A minor chord for guitar (open).svg\|**A minor** File: D minor
chord for guitar (open).svg\|**D minor** File: E minor chord for guitar
(open).svg\|**E minor**

```{=html}
</div>
```
## Barre chords

Barre chords use a movable shape that can be shifted up in pitch and
down in pitch along a string to give different keys.

```{=html}
<div style="font-family:'Courier New'">
```
File: B major chord for guitar (barred).png\|**B major** File: B minor
chord for guitar (barred).png\|**B minor** File: F major chord for
guitar (barred).png\|**F major** File: F minor chord for guitar
(barred).png\|**F minor**

```{=html}
</div>
```
## Open seventh chords

Seventh chords, strictly speaking from a music theory perspective, means
any seventh chord, including major, minor, dominant seventh, diminished
and so on. In a beginning guitar context, however, \"seventh chord\"
often means a dominant seventh chord, which is a major triad with a
flatted seventh. The G dominant seventh chord, or G7 chord, is the notes
G, B, D and F.

```{=html}
<div style="font-family:'Courier New'">
```
File: D7 chord for guitar (open).png\|**D7** File: G7 chord for guitar
(open).png\|**G7** File: A7 chord for guitar (open).png\|**A7** File: C7
chord for guitar (open).png\|**C7**

```{=html}
</div>
```
## Chords in C major

\

```{=html}
<div style="font-family:'Courier New'">
```
File: C major chord for guitar (open).png\|**C major** File: D minor
chord for guitar (open).png\|**D minor** File: E minor chord for guitar
(open).png\|**E minor** File: F major chord for guitar (open).png\|**F
major** File: G major chord for guitar (open).png\|**G major** File: A
minor chord for guitar (open).png\|**A minor** <File:B> half diminished
7th chord for guitar (root 1st position).png\|**B halfdim7**

```{=html}
</div>
```
Note that the chords in the key of C major consists of 3 major chords, 3
minor chords and 1 diminished chord. This holds true for all major keys.

fr:Apprendre la
guitare/Accords


# Guitar/Chords

A chord is two or more different notes played simultaneously. Most
chords have three notes though. Chords derive their names from the
*root* note; so a C chord has C for its root note and a G7 chord will
have G. The interval between the root note and the third determines
whether a chord is a major or minor. Chords may be strummed or the notes
picked individually though beginners find strumming much easier. The
more advanced technique of picking is examined in the Picking and
Plucking chapter.

While chords are primarily used for rhythm guitar, basic chord knowledge
can be important for lead playing as well. Knowing how chords are
constructed can help when learning the lead parts of many songs since
there is always a relationship between a chord and the lead part. For
example, if you have to play a lead part over a C major chord (C-E-G)
and you use the notes of a D flat major chord (Db-F-Ab) then the result
will be very dissonant. Additionally, many lead patterns revolve around
arpeggios. These are chords with their notes played in sequence (the
word \"arpeggio\" actually means \"broken chord\") rather than together.
For more information on arpeggios, see the Arpeggio and Sweep
Picking chapter.

Chords are easy to play though understanding the theory behind chord
construction (harmony) will require some understanding of scales. While
it is not essential to have a knowledge of scales to be able to use this
section, understanding scales will definitely improve your general
musicianship. With that in mind, go ahead and learn and use these chords
without worrying too much about the theory and when you have the time
take a look at the page on general music theory and
the page on scales.

Beginners are advised to start with open
chords, which are often the easiest
chords to form. Learning open chords is important because it sets the
stage for learning how to form barre chords. Barre
Chords are chords you form by pressing
all (or some) of the strings down with the first finger. This finger
acts as the barre (the same job that the nut of the guitar does when you
are playing open chords). Because of this barre chords don\'t usually
include open strings and can be moved freely up and down the neck. As
you move your barre chord, the shape of the chord remains the same
although all the notes change. Barring is an important technique and
greatly opens up the neck of the instrument.

## Different Kinds of Chords

### Major chords

The most basic chord is called a triad and consists of three different
notes. A major triad consists of the root, a major third, and a perfect
fifth. The early study of chords should be based around how to build the
tonic triad (chord) from any major scale. To build the tonic triad you
take the first note of any major scale and the third note (a major 3rd)
and the fifth note (a perfect fifth). Take for example building a tonic
triad (chord) from the C major scale. If you look at this C major scale:

**C-D-E-F-G-A-B-C**

you will notice that the first, third and fifth notes of the scale are
C, E, and G. The most obvious thing that most guitarists become
instantly aware of is that the C major played in the first position
actually involves playing 5 strings and therefore must have more notes
than a triad. The C major chord shown below has these notes C, E, G, C,
E. If you cancel out the doubles you are left with a C major triad. This
brings us to an important rule: any chord tone (note) can be doubled
without affecting the chords designation. Therefore a C major triad
(C-E-G) and the first position C major chord below (C-E-G-C-E) are both
still C major chords though the 5 note version will sound fuller due to
the note doubling. Major chords have a characteristically bright and
happy sound.

```{=html}
<div style="font-family:'Courier New'">
```
File: C major chord for guitar (open).png\|**C major**

```{=html}
</div>
```
### Minor chords

The minor triad consists of a root, a minor third and a perfect fifth.
The interval between the root and third is a minor third and the
interval between the root and fifth is a perfect fifth. Minor chords
have a dissonant quality due to the interval of a minor third. It must
be remembered that we are talking about building chords from scales and
that these intervals, the minor third and perfect fifth, are the
interval designations from the scale which are then applied to naming
the intervals in a chord. Which is why the triad intervals are not named
1st, 2nd and 3rd respectively.

Minor chords are best understood in relation to their major chord
counterpart. In the example below we will use E major and E minor. When
we play an E major chord, we can flatten the third of the chord by
lifting the finger that is holding down the third string at the first
fret, making it an open string. By altering this one note so that the
interval is changed from a major third to a minor third, we have formed
a new chord: E minor.

```{=html}
<div style="font-family:'Courier New'">
```
File: E major chord for guitar (open).png\|**E major** File: E minor
chord for guitar (open).png\|**E minor**

```{=html}
</div>
```
Switching between major and minor chords can be relatively easy, as it
involves the change of only one note. Some chord changes, for example
changing between an open F major to a F minor, will need a little more
effort.

### Dominant Seventh chords

A minor seventh is added to a major chord. When a minor seventh is added
to any major chord that major chord is changed into a dominant seventh.
The dominant chord always refers to the chord built on the fifth degree
of any major scale. Look at the C major scale below:

**C-D-E-F-G-A-B-C**

The fifth degree of the scale is G. The chord built on the fifth
contains the notes: G-B-D. To change this dominant major chord to a
dominant seventh you need to add a fourth note. The note you add is F
(the minor seventh) which now makes: G-B-D-F. This chord has very strong
need to resolve usually to the tonic. The reason the interval in G-B-D-F
is called a minor seventh and not a perfect fourth is that interval
designation is determined from the root of the chord being discussed.
Take for example the G major scale below:

**G-A-B-C-D-E-F#-G**

As you can see the interval G-F# is a major seventh. You can form a
minor seventh interval by lowering the seventh by a semitone: G-F. This
holds true for all major seventh intervals.

At first it seems quite strange looking at the interval relationship of
another key to determine the chord intervals of the key you are playing
in. With practice it becomes very easy but does involve learning a few
major scales.

```{=html}
<div style="font-family:'Courier New'">
```
<File:A7> chord for guitar (open).png\|**A7** <File:G7> chord for guitar
(open).png\|**G7**

```{=html}
</div>
```
### Sixth chords

Add a sixth to the chord. The two chords below are major chords from the
key of C with a sixth added.

```{=html}
<div style="font-family:'Courier New'">
```
<File:F6> chord for guitar (open position).png\|**F6 (Subdominant in C
major)** <File:C6> chord for guitar (open position).png\|**C6 (Tonic in
C major)**

```{=html}
</div>
```
### Suspended chords

To form a suspended chord the third is replaced with either a second or
a fourth. The third of a chord defines its modality - whether a chord is
major or minor. By removing the third and replacing it with a second or
fourth you have suspended the chord\'s modal quality. This creates a
chord that is neither major or minor and the ear interprets the chord as
harmonically ambiguous.

Suspended chords derived from a D major chord:

```{=html}
<div style="font-family:'Courier New'">
```
<File:Dsus2> chord for guitar (open position).png\|**Dsus2** File: D
major chord for guitar (open).png\|**D major** <File:Dsus4> chord for
guitar (open position).png\|**Dsus4**

```{=html}
</div>
```
Suspended chords derived from the A major chord:

```{=html}
<div style="font-family:'Courier New'">
```
<File:Asus2> chord for guitar (open position).png\|**Asus2** File: A
major chord for guitar (open).png\|**A major** <File:Asus4> chord for
guitar (open position).png\|**Asus4**

```{=html}
</div>
```
Suspending an E major chord:

```{=html}
<div style="font-family:'Courier New'">
```
File: E major chord for guitar (open).png\|**E major** <File:Esus4>
chord for guitar (open position).png\|**Esus4**

```{=html}
</div>
```
### Slash chords

Chords that are not in root position. For example, a C/G is a C chord
with a bass note of G. They are also referred to as \"inversions\".
Slash chords are always notated with their chord name first followed by
the bass note.

```{=html}
<div style="font-family:'Courier New'">
```
<File:C> major chord for guitar (open position in second
inversion).png\|**C/G** <File:F> major chord for guitar (open position
in second inversion).png\|**F/C**

```{=html}
</div>
```
### Diminished chords

These consist of a stack of minor thirds. You can extend a diminished
triad (three note chord) by adding another minor third; which gives you
a four note chord called a diminished seventh chord. The diminished
seventh chord is notated as Co7 or dim7. Diminished seventh chords are
built entirely from minor thirds, so you can move the chord shape up the
neck in intervals of a minor third (three frets) and this will be
exactly the same notes as the original chord but in a different order.
The term \"inversion\" is used when chords have their notes rearranged.

```{=html}
<div style="font-family:'Courier New'">
```
<File:Edim7> chord for guitar (root).png\|**Edim7 - root position**
<File:Edim7> chord for guitar (third in bass).png\|**Edim7 - 1st
inversion** <File:Edim7> chord for guitar (fifth in bass).png\|**Edim7 -
2nd inversion**

```{=html}
</div>
```
A half-diminished chord consists of a diminished triad with a major
third on top. In other words, a half-diminished chord is a diminished
triad with a minor seventh.

Diminished chords are full of tension because of the dissonance created
by stacking minor third intervals and they are normally resolved to a
consonant major or minor chord.


# Timeless Theorems of Mathematics/Polynomial Remainder Theorem

The **Polynomial Remainder Theorem** is an application of Euclidean
division of polynomials. It is one of the most fundamental and popular
theorems of Algebra. It states that the remainder of the division of a
polynomial $P(x)$ by a linear polynomial $(x - r)$ is equal to $P(r)$.

## Examples

### Example 1

Show that the remainder of the division of a polynomial
$f(x) = x^2 - 2x + 2$ by a linear polynomial $(x - 1)$ is equal to
$f(1)$. **Solution :** Divide $f(x) = x^2 - 2x + 2$ by $(x - 1)$ like
the following one.

``` cpp
x - 1 ) x^2 - 2x + 2 ( x - 1
        x^2 - x
        ------------
            - x + 2
            - x + 1
        ------------
                  1
```

As, $f(1) = 1^2 - 2(1) + 2 = 1$, thus the remainder is equal to $f(1)$.

### Example 2

Show that the remainder of the division of a polynomial
$F(x) = ax^2 + bx + c$ by a linear polynomial $(x - m)$ is equal to
$F(m)$. **Solution :** Divide $F(x) = ax^2 + bx + c$ by $(x - m)$ like
the following one.

``` cpp
x-m ) ax^2+bx+c ( ax+am+b
      ax^2-amx
      ------------------
           amx+bx+c
           amx     -am^2
      ------------------
               bx+c+am^2
               bx-bm
      ------------------
               am^2+bm+c
```

As, $f(m) = am^2 + bm + c$, thus the remainder $am^2 + bm + c$ is equal
to $f(m)$.

## Proof

### Proposition

If $P(x)$ is a polynomial of a positive degree and $a$ is any definite
number, the remainder of the division of $P(x)$ by $(x - a)$ will be
$P(a)$

### Proof

The remainder of the division of a polynomial of a positive degree
$P(x)$ by $(x - a)$ is either 0 or a non-zero constant. Let the
remainder is $R$ and the quotient is $Q(x)$. Then, for every value of
$x$,

```{=html}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
```
$P(x) = (x - a)$·$Q(x) + R .... (i)$

```{=html}
</div>
```
Putting $x = a$ in the equation $(i)$, we get $P(a) = 0$ ·
$Q(a) + R = R$. Thus, the remainder of $P(x)$÷$(x - a)$ is equal to
$P(a)$.


# Timeless Theorems of Mathematics/Polynomial Factor Theorem

The **Polynomial Factor Theorem** is a theorem linking factors and zeros
of a polynomial.[^1] It is an application of the polynomial remainder
theorem Polynomial Remainder
Theorem.
It states that a polynomial $f(x)$ has a factor $(x - a)$ if and only if
$f(a) = 0$. Here, $a$ is also called the root of the polynomial.[^2]

## Proof

### Statement

If $P(x)$ is a polynomial of a positive degree and if $P(a) = 0$ so
$(x - a)$ is a factor of $P(x)$.

### Proof

According to the Polynomial Remainder Theorem, the remainder of the
division of $P(x)$ by $(x - a)$ is equal to $P(a)$. As $P(a) = 0$, so
the polynomial $P(a)$ is divisible by $(x - a)$

∴ $(x - a)$ is a factor of $P(x)$. *\[Proved\]*

### Converse of Factor Theorem

**Proposition :** If $(x - a)$ is a factor of the polynomial $P(x),$
then $P(a) = 0$

## Factorization

### Example 1

**Problem :** Resolve the polynomial $P(x) = 18x^3 + 15x^2 - x - 2$ into
factors.

**Solution :** Here, the constant term of $P(x)$ is $-2$ and the set of
the factors of $-2$ is $F$**~1~**$=${±1, ±2}

Here, the leading coefficient of $P(x)$ is $18$ and the set of the
factors of $18$ is $F$**~2~**$=${±1, ±2, ±3, ±6, ±9, ±18}

Now consider $P(a)$, where
$a = \frac {r}{s}, r\isin F$**~1~**$, s\isin F$**~2~**

When,

$a = 1, P(1) = 18+15-1-2 \ne 0$

$a = -1, P(-1) = -18+15+1-2 \ne 0$

$a = -\frac{1}{2}, P(-\frac{1}{2}) = 18(-\frac{1}{8})+15(-\frac{1}{4})+\frac{1}{2}-2 = 0$

Therefore, $(x + \frac{1}{2}) = \frac{1}{2}(2x+1)$ is a factor of $P(x)$

Now, $P(x) = 18x^3 + 15x^2 - x - 2$
$= 18x^3 + 9x^2 + 6x^2 + 3x - 4x - 2$
$= 9x^2(2x + 1) + 3x(2x + 1) - 2(2x + 1)$ $= (2x + 1)(9x^2 + 3x - 2)$
$= (2x + 1)(9x^2 + 6x - 3x - 2)$ $= (2x + 1)(3x(3x + 2) - 1(3x + 2))$
$= (2x + 1)(3x + 2)(3x - 1)$

∴$P(x) = (2x + 1)(3x + 2)(3x - 1)$

### Example 2

**Problem :** Resolve the polynomial
$P(x) = -3x^2 -2xy + 8y^2 + 11x - 8y - 6$ into factors.

**Solution :** Considering only the terms of $x$ and constant, we get
$-3x^2+ 11x - 6$.

$-3x^2+ 11x - 6 \equiv (-3x+2)(x-3)..(i)$

In the same way, considering only the terms of $y$ and constant, we get
$8y^2 - 8y - 6$.

$8y^2 - 8y - 6 \equiv (4y+2)(2y-3)..(ii)$

Combining factors of above (i) and (ii), the factors of the given
polynomial can be found. But the constants $+2, -3$ must remain same in
both equations just like the coefficients of $x$ and $y$.

∴$P(x) = (-3x+4y+2)(x+2y-3)$

## References

[^1]: 1 Byjus.com, Maths,
    Factor Theorem

[^2]: 2 Byjus.com,
    Maths, Roots of Polynomials


# Timeless Theorems of Mathematics/Rational Root Theorem

The rational root theorem states that, if a rational number
$\frac{p}{q}$ (where $p$ and $q$ are relatively prime) is a root of a
polynomial with integer coefficients, then $p$ is a factor of the
constant term and $q$ is a factor of the leading coefficient. In other
words, for the polynomial,
$P(x) = a_{n}x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ...+ a_0$, if
$P(\frac{p}{q}) = 0$, (where $a_i\in\Z$ and $a_0, a_n \neq 0$) then
$\frac{(a_0)}{p}\in\Z$ and $\frac{a_n}{q}\in\Z$

## Proof

Let $P(x) = a_{n}x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ...+ a_0$,
where $a_i\in\Z$.

Assume $P(\frac{p}{q}) = 0$ for coprime $p, q\in\Z$. Therefore,
$P(\frac{p}{q})=a_n(\frac{p}{q})^n+a_{n-1}(\frac{p}{q})^{n-1} + a_{n-2}(\frac{p}{q})^{n-2}+...+a_1(\frac{p}{q})+a_0 = 0$
$\Rightarrow a_np^n + a_{n-1}p^{n-1}q+a_{n-2}p^{n-2}q^2+...+a_1pq^{n-1}+a_0q^n=0$
$\Rightarrow p(a_np^n-1 + a_{n-1}p^{n-2}q+a_{n-3}p^{n-2}q^2+...+a_1q^{n-1})=-a_0q^n$

Let $w=a_np^n-1 + a_{n-1}p^{n-2}q+a_{n-3}p^{n-2}q^2+...+a_1q^{n-1}\in\Z$

Thus, $w=-\frac{a_0q^n}{p}$

As $p$ is coprime to $q$ and $w\in\Z.$, thus $\frac{a_0}{p}\in\Z$.

Again,
$a_np^n + a_{n-1}p^{n-1}q+a_{n-2}p^{n-2}q^2+...+a_1pq^{n-1}+a_0q^n=0$
$\Rightarrow q(a_{n-1}p^{n-1}+a_{n-2}p^{n-2}q+...+a_1pq^{n-2}+a_0q^{n-1})=-a_np^n$

Let $(a_{n-1}p^{n-1}+a_{n-2}p^{n-2}q+...+a_1pq^{n-2}+a_0q^{n-1})=v\in\Z$

Thus, $qv=-a_np^n$ $\Rightarrow v=-\frac{a_np^n}{q}$

As $q$ is coprime to $p$ and $v\in\Z.$, thus $\frac{a_n}{p}\in\Z$.

$\therefore$ For
$P(x) = a_{n}x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ...+ a_0$, if
$P(\frac{p}{q}) = 0$, (where $a_i\in\Z$ and $a_0, a_n \neq 0$) then
$\frac{(a_0)}{p}\in\Z$ and $\frac{a_n}{q}\in\Z$. ***\[Proved\]***


# Timeless Theorems of Mathematics/Binomial Theorem

The Binomial Theorem is a fundamental theorem in algebra that provides a
formula to expand powers of binomials. It allows us to easily expand
expressions, $(a+b)^n$, where $a$ and $b$ are real numbers or variables,
and $n\geq0$.

## Proof

**Proposition:** For any real numbers $a, b$ and $n\ge0,$
$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k$ Where
$\binom{n}{k} = \frac{n!}{k! (n-k)!}$

**Proof *(Mathematical Induction)*:**

For $n = 0, (a + b)^n = (a + b)^0 = 1$

For $n = 1, (a + b)^n = (a + b)^1 = (a + b)$

For $n = 2, (a + b)^n = (a + b)^2$ $= a^2 + 2ab + b^2$
$= \binom{2}{0}a^2b^0 + \binom{2}{1}a^1b^1 + \binom{2}{2}a^0b^2$
$=\sum_{k=0}^2 \binom{2}{k}a^{2-k}b^k$

For $n = 3, (a + b)^n = (a + b)^3$ $= a^3 + 3a^2b + 3ab^2 + b^3$
$= \binom{3}{0}a^3b^0 + \binom{3}{1}a^2b^1 + \binom{3}{2}a^1b^2 + \binom{3}{3} a^0b^3$
$=\sum_{k=0}^3 \binom{3}{k}a^{3-k}b^k$

Let\'s assume $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k$ for
some $a\ge0.$ Now we just have to show that the equation holds true for
$n + 1.$

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k$

Or, $(a + b)^n(a + b) = (\sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k)(a + b)$

$= (\binom {n}{0} a^nb^0 + \binom {n}{1} a^{n-1}b^1 + \binom {n}{2} a^{n-2}b^2 + ... + \binom {n}{n} a^0b^n) (a + b)$

$= a(\binom {n}{0} a^nb^0 + \binom {n}{1} a^{n-1}b^1 + \binom {n}{2} a^{n-2}b^2 + ... + \binom {n}{n} a^0b^n) + b(\binom {n}{0} a^nb^0 + \binom {n}{1} a^{n-1}b^1 + \binom {n}{2} a^{n-2}b^2 + ... + \binom {n}{n} a^0b^n)$

$= (\binom {n}{0} a^{n+1}b^0 + \binom {n}{1} a^{n}b^1 + \binom {n}{2} a^{n-1}b^2 + ... + \binom {n}{n} ab^n) + (\binom {n}{0} a^nb^1 + \binom {n}{1} a^{n-1}b^2 + \binom {n}{2} a^{n-2}b^3 + ... + \binom {n}{n} a^0b^{n+1})$

$= \binom {n}{0} a^{n+1}b^0 + a^{n}b^1 (\binom {n}{1} + \binom {n}{0}) + a^{n-1}b^2 (\binom {n}{2} + \binom {n}{1}) + ... + ab^n (\binom {n}{n} + \binom {n}{n-1}) + \binom {n}{n} a^0b^{n+1})$

$= \binom {n+1}{0} a^{n+1}b^0 + \binom {n+1}{1}a^{n}b^1 + \binom {n+1}{2} a^{n-1}b^2 + ... + \binom {n + 1}{n} ab^n + \binom {n+1}{n+1} a^0b^{n+1})$

∴$(a + b)^{n+1} = \sum_{k=0}^{n+1} \binom{n+1}{k}a^{n+1-k}b^k$


# Timeless Theorems of Mathematics/Product, Quotient, Composition Rules

The product rule, the quotient rule and the composition *(or chain)*
rule are the most fundamental rules or formulas of differential
calculus. For the functions $u(x)$ and $v(x)$, the rules are,

1.  Product Rule: $D_x (uv) = v\cdot D_x u + u\cdot D_x v$
2.  Quotient Rule:
    $D_x (\frac{u}{v}) = \frac{1}{v^2} (v \cdot D_x u - u \cdot D_x v)$
3.  Composition Rule: $D_x v = D_y v\cdot D_x y$

## Proof

### Product Rule

$D_x (uv)$ $=\lim_{{\Delta x \to 0}} \frac {\Delta (uv)}{\Delta x}$
$=\lim_{{\Delta x \to 0}} \frac{u(x+\Delta x)\cdot v(x+\Delta x) - u(x)\cdot v(x)}{\Delta x}$
$=\lim_{{\Delta x \to 0}} \frac{u(x+\Delta x)\cdot v(x+\Delta x) - u(x)\cdot v(x+\Delta x) + u(x)\cdot v(x+\Delta x) - u(x)\cdot v(x)}{\Delta x}$
$= \lim_{{\Delta x \to 0}} \frac{[u(x+\Delta x)-u(x)]\cdot v(x+\Delta x) + u(x)\cdot [v(x+\Delta x)-v(x)]}{\Delta x}$
$=\lim_{{\Delta x \to 0}} \frac{(\Delta u) \cdot v(x+\Delta x) + u(x)\cdot (\Delta v)}{\Delta x}$
$= \lim_{{\Delta x \to 0}}[\frac {\Delta u}{\Delta x}\cdot v(x + \Delta x) + \frac {\Delta v}{\Delta x}\cdot u(x)]$
$= \lim_{{\Delta x \to 0}}[D_x u\cdot v(x + \Delta x) + u\cdot D_x v]$
$=v\cdot D_x u + u\cdot D_x v$

$\therefore D_x (uv) = v\cdot D_x u + u\cdot D_x v$ ***\[Proved\]***

### Quotient Rule

$D_x (\frac{u}{v})$
$=\lim_{{\Delta x \to 0}} \frac {\Delta (\frac {u}{v})}{\Delta x}$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)}{v(x + \Delta x)} - \frac {u(x)}{v(x)}]$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)\cdot v(x) - u(x)\cdot v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)\cdot v(x) -v(x)\cdot u(x) + v(x)\cdot u(x) - u(x)\cdot v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {v(x)\cdot (u(x + \Delta x)-u(x)) + u(x)\cdot (v(x) - v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$
$=\lim_{{\Delta x \to 0}} \frac {1}{\Delta x}\cdot [\frac {v(x)\cdot (\Delta u) + u(x)\cdot (\Delta v)}{v(x)\cdot v(x + \Delta x)}]$
$=\lim_{{\Delta x \to 0}} \frac {\frac {v(x)\cdot (\Delta u)}{\Delta x} + \frac {u(x)\cdot (\Delta v)}{\Delta x}}{v(x)\cdot v(x + \Delta x)}$
$=\lim_{{\Delta x \to 0}} \frac{{v(x) \cdot \frac{\Delta u}{\Delta x} + u(x) \cdot \frac{\Delta v}{\Delta x}}}{{v(x) \cdot v(x + \Delta x)}}$
$=\lim_{{\Delta x \to 0}} \frac{{v(x) \cdot D_x u + u(x) \cdot D_x v}}{{v(x) \cdot v(x + \Delta x)}}$
$=\frac{{v(x) \cdot D_x u + u(x) \cdot D_x
v}}{{v(x) \cdot v(x)}}$ $=\frac{{v\cdot D_x u + u\cdot D_x v}}{{v^2}}$

$\therefore D_x (\frac{u}{v}) = \frac{v \cdot D_x u - u \cdot D_x v}{v^2}$

### Composition Rule

$D_xv$ $= \lim_{\Delta x\to 0}\frac{\Delta v}{\Delta x}$
$= \lim_{\Delta x\to 0}\frac{\Delta v\cdot\Delta y}{\Delta x\cdot\Delta y}$
$= \lim_{\Delta x\to 0}\frac{\Delta v}{\Delta y}\cdot\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}$
$= \lim_{\Delta y\to 0}\frac{\Delta v}{\Delta y}\cdot\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}$
$= D_yv\cdot D_xy$

$\therefore D_xv = D_yv\cdot D_xy$


# Timeless Theorems of Mathematics/Intermediate Value Theorem

The **Intermediate Value Theorem** is a fundamental theorem in calculus.
The theorem states that if a function, $f(x)$ is continuous on a closed
interval $[a,b],$ then for any value, $y$ defined between $(f(a)$ and
$f(b),$ there exists at least one value $c \in (a,b)$ such that
$f(c) = y$. ![Intermediate Value Theorem: f(x) is continuous on \[a,
b\], there exists at least one value c, that is defined on (a, b) such
that f(c) =
y.](Intermediatevaluetheorem.svg "Intermediate Value Theorem: f(x) is continuous on [a, b], there exists at least one value c, that is defined on (a, b) such that f(c) = y.")

## Proof

**Statement:** If a function, $f$ is continuous on $[a, b],$ then for
every $y$ between $f(a)$ and $f(b),$ there exists at least one value
$c \in (a, b)$ such that $f(c) = y$

**Proof:** Assume that $f(x)$ is a continuous function on $[a, b]$ and
$f(a)<f(b).$

Consider a function $g(x) = f(x) - y.$ The purpose of defining $g(x)$ is
to investigate the behavior of $f(x)$ concerning the value $y$.

Since $f$ is continuous on $[a,b]$ and $y$ is a constant,
$g(x) = f(x) - y$ is also continuous on $[a,b],$ as the difference of
two continuous functions is continuous.

Now, $f(a) < y$ \[As $c \in (a, b)$ and $y = f(c)$\]

Or, $f(a) - y < 0$

$\therefore g(a) < 0$

In the same way, $g(b) > 0$

Since $g(x)$ is continuous and $g(a)$ is defined below the $x$-axis
while $g(b)$ is defined above the $x$-axis, there must exist at least
one point $c$ in the interval $[a,b]$ where $g(c)=0$.

Therefore, at the point $c$, $g(c) = f(c) - y = 0 \implies f(c) = y$

∴ There exists at least one point $c$ in the interval $[a,b]$ such that
$f(c)=y.$ ***\[Proved\]***


# Timeless Theorems of Mathematics/Differentiability Implies Continuity

**Statement:** If a function $f(x)$ is differentiable at the point
$x_0,$ then $f$ is continuous at $x_0.$

**Proof:** Assume $f(x)$ is differentiable at $x_0$. Then, $D_{x}f(x_0)$
exists.

The function $f(x)$ is continuous at $x_0$ when
$\lim_{x \to x_0} f(x) = f(x_0).$
$\Rightarrow \lim_{x \to x_0} f(x) - f(x_0) = 0$
$\Rightarrow \lim_{x \to x_0} f(x) - \lim_{x \to x_0} f(x_0) = 0$
$\Rightarrow \lim_{x \to x_0} [f(x) - f(x_0)] = 0.$ So, to prove the
theorem, it is enough to prove that
$\lim_{x \to x_0} [f(x) - f(x_0)] = 0.$

$\lim_{x \to x_0} [f(x) - f(x_0)]$
$=\lim_{x \to x_0} [\frac {f(x) - f(x_0)}{x - x_0} (x-x_0)]$
$=\lim_{x \to x_0} [\frac {f(x) - f(x_0)}{x - x_0}]\cdot\lim_{x \to x_0} (x-x_0)$
$=D_{x}f(x_0)\cdot 0 = 0$

Therefore, When $D_{x}f(x_0)$ exists,
$\lim_{x \to x_0} [f(x) - f(x_0)] = 0$ is true.

$\therefore$ If $f(x)$ is differentiable at the point $x_0,$ then $f$ is
continuous at $x_0.$ ***\[Proved\]***


# Timeless Theorems of Mathematics/Rolle's Theorem

The Rolle\'s theorem says that, If a real-valued function $f$ is
continuous on a closed interval $[a, b]$, differentiable on an open
interval $(a, b)$ and $f(a) = f(b)$, then there exists at least a number
$c$ such that $D_xf(c) = 0$. It means that if a function satisfies the
three conditions mentioned in the previous sentence, then there is at
least a point in the graph of the function, where the slope of the
tangent line at the point is $0$, or the tangent line is parallel to the
$x$-axis.

## Proof

![f(x) is continuous on $[a, b]$ differentiable on $(a, b)$ and
$f(a)=f(b)$. Thus,
$D_xf(c)=0$](RTCalc.svg "f(x) is continuous on [a, b] differentiable on (a, b) and f(a)=f(b). Thus, D_xf(c)=0")


# Timeless Theorems of Mathematics/Bézout's Identity

!Étienne Bézout *(31 March 1730 -- 27 September
1783)*")
**Bézout\'s Identity** is a theorem of Number Theory and Algebra, which
is named after the French mathematician, Étienne Bézout *(31 March 1730
-- 27 September 1783)*. The theorem states that the greatest common
divisor, $$ of the integers, $a$ and $b$ can be written in the form,
$ax + by = d,$ where $x$ and $y$ are integers. Here, $x$ and $y$ are
called Bézout coefficients for $(a, b)$.

## Computing the pairs, $(x, y)$

There are infinite number of pairs of $(x, y)$ which satisfies the
equation $ax + by = d,$. A general formula can be developed to compute
pairs as much as you want. To do that, first of all, it\'s required to
calculate one pair of $(x, y)$. One simple way to calculate a pair is
using the **extended Euclidean algorithm**.

### General Formula for Computing

Once you have one pair $(x_0, y_0),$ you can apply the formula,
$(x_k, y_k) = (x_0 - k\frac{b}{d}, y_0 + k\frac{a}{d})$where $k\in \Z$,
that means $k$ is an integer.

**Proof:** As $(x, y)=(x_0, y_0)$ satisfies the equation $ax + by = d,$
then,

$ax_0 + by_0 = d$

Or, $\frac {d(ax_0 + by_0)}{d} = d$

Or, $\frac {dax_0 + dby_0}{d} = d$

Or, $\frac {dax_0 -kba + kba + dby_0}{d} = d$

Or, $ax_0 - ak\frac{b}{d} + by_0 + bk\frac{a}{d} = d$

Or, $a(x_0 - k\frac{b}{d}) + b(y_0 + k\frac{a}{d}) = d$

Or, $a(x_0 - k\frac{b}{d}) + b(y_0 + k\frac{a}{d}) = ax_k + by_k$

Therefore, the coefficients of $a$ are equal and the coefficients of $b$
are also equal, $(x_k, y_k) = (x_0 - k\frac{b}{d}, y_0 + k\frac{a}{d})$

*\[Note: The formula only works when $d \neq 0$. Also, as $x\in\Z$, then
$k\in\Z$.\]*

**Example:** The greatest common divisor of $a = 8$ and $b = 12$ is
$\gcd(8, 12) = 4.$ According to the identity, there exists integers $x$
and $y$, so that $8x + 12y = 4$ $\Rightarrow 2x + 3y = 1$. If you try to
solve the equation, you may soon come up with a pair of solutions like
$(x_0, y_0) = (-1, 1)$. So,
$(x_k, y_k) = (-(1 + k\frac{b}{d}), 1 + k\frac{a}{d})$. By using this
formula, you may find pairs as much as you want.

## Proof

Assume $S = \{ax+by:x,y\in\Z \text{ and } ax+by>0\}$ where $a$ and $b$
are non zero integers. The set is not an empty set as it contains either
$a$ or $-a$ when $x = \pm 1$ and $y=0$. Since $S$ is not an empty set,
by the well-ordering principle, the set has a minimum element
$d = as + bt$.

The Euclidean division for $\frac {a}{d}$ may be written as
$a = dq + r$, where $q=\left\lfloor {\frac{a}{d}} \right\rfloor$,
$r = a - d \left\lfloor {\frac{a}{d}} \right\rfloor$ and $0 \leq r <d$.

Here, $r = a - dq$ $= a - q(as + bt)$ $= a - qas + qbt$
$= a(1 - qs) + b(qt)$

Thus, $r$ is in the form $ax+by$, and hence $r\in S\cup\{0\}$. But,
$0\leq r<d$ and $d$ is the smallest positive integer in $S$. So, $r$
must be $0$. Thus, $d$ is a divisor of $a$. $q = \frac {a}{d} > 0$ as
the remainder is zero and a is a non zero integer. Similarly, $d$ is
also a divisor of $b$. Therefore, $d$ is a common divisor of $a$ and
$b$.

Assume $c$ as any common divisor of $a$ and $b$; $a = cu$, $b = cv$.
Again, $d = as+bt$ $= cus + cvt$ $= c(us + vt)$

Thus, $c$ is a divisor of d. Since $d>0$ $\implies c\leq d$. Therefore,
any common divisor of $a$ and $b$ is less than or equals to $d$.

$\therefore d$ is the same as $gcd(a, b)$ and $d$ can be expressed as
$ax + by = d$. ***\[Proved\]***


# Timeless Theorems of Mathematics/De Morgan's laws

**De Morgan\'s Law** is a fundamental principle in Logic and Set Theory.
It establishes a useful relationship between the logical operators
\'AND\' and \'OR\' when negations *(NOT)* are applied. There are two
primary forms of De Morgan\'s Law, known as De Morgan\'s First Law and
De Morgan\'s Second Law. These laws state that,

1.  The negation of a disjunction is the conjunction of the negations,
    $(A \cup B)' = A' \cap B'$
2.  The negation of a conjunction is the disjunction of the negations,
    $(A \cap B)' = A' \cup B'$

## Proof

!De Morgan\'s Law for sets $A$ and
$B$ Let $A$ and
$B$ are two sets. De Morgan\'s First Law states that the complement of
the union of $A$ and $B$ sets is the same as the intersection of the
complements of $A$ and $B$. That means, $(A \cup B)' = A' \cap B'$, or
$\overline {(A\cdot B)} = (\overline {A} + \overline {B})$. And De
Morgan\'s First Law states that the complement of the intersection of
$A$ and $B$ is the same as the union of their complements,
$(A \cap B)' = A' \cup B'$, or
$\overline {(A+B)} = (\overline {A} \cdot \overline {B})$.

### De Morgan\'s First Law

Assume $x \in (A\cup B)'$. Then $x \not \in A \cup B$.

$\implies x \not \in A$ AND $x \not \in B$

$\implies x \in A'$ AND $x \in B'$

$\implies x \in (A'\cap B')$

$\therefore (A\cup B)' \subseteq (A'\cap B')$

Again, Let, $x \in (A'\cap B')$. Then, $x \in A'$ AND $x \in B'$

$\implies x \not \in A$ AND $x \not \in B$

$\implies x \not \in A \cup B$.

$\implies x \in (A\cup B)'$

$\therefore (A' \cap B') \subseteq (A\cup B)'$

Therefore, $(A \cap B)' = A' \cup B'$ ***\[Proved\]***

### De Morgan\'s Second Law

Assume $x \in (A\cap B)'$. Then $x \not \in A \cap B$.

$\implies x \not \in A$ OR $x \not \in B$

$\implies x \in A'$ OR $x \in B'$

$\implies x \in (A'\cup B')$

$\therefore (A\cap B)' \subseteq (A'\cup B')$

Again, Let, $x \in (A'\cup B')$. Then, $x \in A'$ OR $x \in B'$

$\implies x \not \in A$ OR $x \not \in B$

$\implies x \not \in A \cap B$.

$\implies x \in (A\cap B)'$

$\therefore (A' \cup B') \subseteq (A\cap B)'$

Therefore, $(A \cap B)' = A' \cup B'$ ***\[Proved\]***


# Timeless Theorems of Mathematics/Mid Point Theorem

!In this right triangle, $AD = CD, BE = CE, DE = \frac {1}{2}AB$and
$DE \parallel AB$ according to the Mid Point
Theorem
The midpoint theorem is a fundamental concept in geometry that
establishes a relationship between the midpoints of a triangle\'s sides.
This theorem states that when you connect the midpoints of two sides of
a triangle, the resulting line segment is parallel to the third side.
Additionally, this line segment is precisely half the length of the
third side.

## Proof

### Statement

In a triangle, if a line segment connects the midpoints of two sides,
then this line segment is parallel to the third side and half its
length.

### Proof with the help of Congruent Triangles

!The construction for the mid-point theorem\'s proof with similar
triangles
**Proposition:** Let $D$ and $E$ be the midpoints of $AC$ and $BC$ in
the triangle $ABC$. It is to be proved that,

1.  $DE \parallel AB$ and;
2.  $DE = \frac {1}{2}AB$.

**Construction:** Add $D$ and $E$, extend $DE$ to $F$ as $EF = DE$, and
add $B$ and $F$.

**Proof:** ***\[1\]*** In the triangles $\Delta CDE$ and $\Delta EBF,$

$CE = BE$ ; \[Given\]

$DE = EF$ ; \[According to the construction\]

$\angle CED = \angle AEF$ ; \[Vertical Angles\]

∴ $\Delta CDE \cong \Delta EBF$ ; \[Side-Angle-Side theorem\]

So, $\angle CDE = \angle BFE$

∴ $CD \parallel BF$

Or, $AD \parallel BF$ and $CD = BF = DA$

Therefore, $ADFB$ is a parallelogram.

∴ $DF \parallel AB$ or $DE \parallel AB$

***\[2\]*** $DF = AB$

Or $DE + EF = AB$

Or, $DE + DE = AB$ \[As, $\Delta CDE \cong \Delta EBF$\]

Or, $2DE = AB$

Or, $DE = \frac {1}{2}AB$

∴ In the triangle $\Delta ABC,$ $DE \parallel AB$ and
$DE = \frac {1}{2}AB$, where $D$ and $E$ are the midpoints of $AC$ and
$BC$. ***\[Proved\]***

### Proof with the help of Coordinate Geometry

**Proposition:** Let $D$ and $E$ be the midpoints of $AC$ and $AB$ in
the triangle $ABC$, where the coordinates of $A, B, C$ are
$A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)$. It is to be proved that,

1.  $DE = \frac {1}{2}BC$ and
2.  $DE \parallel BC$

**Proof:** ***\[1\]*** The distance of the segment
$BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}$

The midpoint of $A(x_1, y_1)$ and $C(x_3, y_3)$ is
$D(\frac {x_1 + x_3}{2}, \frac {y_1 + y_3}{2})$.

In the same way, The midpoint of $A(x_1, y_1)$ and $B(x_2, y_2)$ is
$E(\frac {x_1 + x_2}{2}, \frac {y_1 + y_2}{2})$

∴ The distance of
$DE = \sqrt{(\frac {x_1 + x_3}{2} - \frac {x_1 + x_2}{2})^2 + (\frac {y_1 + y_3}{2} - \frac {y_1 + y_2}{2})^2}$

$= \sqrt{(\frac {x_1 + x_3 - x_1 - x_2}{2})^2 + (\frac {y_1 + y_3 - y_1 - y_2}{2})^2}$

$= \sqrt{\frac {(x_3 - x_2)^2}{4} + \frac {(y_3 - y_2)^2}{4}}$

$= \sqrt{\frac {(x_3 - x_2)^2 + (y_3 - y_2)^2}{4}}$

$= \frac{\sqrt {(x_3 - x_2)^2 + (y_3 - y_2)^2}}{2}$

$= \frac {1}{2} BC$ ; \[As,
$BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}$\]

***\[2\]*** The slope of $BC,$ $m_1 = \frac {y_2 - y_3}{x_2 - x_3}$

The slope of $DE,$
$m_2 = \frac {\frac {y_1 + y_2}{2} - \frac {y_1 + y_3}{2}}{\frac {x_1 + x_2}{2} - \frac {x_1 + x_3}{2}}$
$= \frac {\frac {y_1 + y_2 - y_1 - y_3}{2}}{\frac {x_1 + x_2 - x_1 - x_3}{2}}$
$= \frac {\frac {y_2 - y_3}{2}}{\frac {x_2 - x_3}{2}}$
$= \frac {y_2 - y_3}{x_2 - x_3}$ $= m_1$ ; \[As,
$m_1 = \frac {y_2 - y_3}{x_2 - x_3}$\]

Therefore, $DE \parallel BC$

∴ In the triangle $\Delta ABC,$ $DE \parallel BC$ and
$DE = \frac {1}{2}BC$, where $D$ and $E$ are the midpoints of $AC$ and
$AB$. ***\[Proved\]***


# Timeless Theorems of Mathematics/Pythagorean Theorem

The Pythagorean Theorem is a fundamental relation between the three
sides of a right triangle in Euclidean Geometry. It states that the
square of the hypotenuse side is equal to the sum of squares of the
other two sides in a right-angled triangle.[^1] !In this right
triangle, $a^2 + b^2 = c^2$ according to the Pythagorean
Theorem
The theorem is named after the Greek philosopher Pythagoras (570 -- 495
BC). But, both the Egyptians and the Babylonians were aware of versions
of the Pythagorean theorem about 1500 years before Pythagoras.

## Proof

### Statement

In a right-angled triangle, the square on the hypotenuse is equal to the
sum of the squares on the two other sides.

### Proof with the help of two right angled triangles

**Proposition :** Let in the triangle $ABC, \angle B = 90$°, the
hypotenuse $AC=b,$ $AB =c$ and $BC= a$. It is required to prove that
$AC^2 = AB^2 + BC^2,$ i.e. $b^2=c^2+a^2$.

**Construction :** Produce $BC$ up to $D$ in such a way that
$CD = AB = c$. Also, draw perpendicular $DE$ at $D$ on $BC$ produced, so
that $DE = BC = a$. Join $C, E$ and $A, E$. !James A. Garfield\'s Proof
of the Pythagorean Theorem using a
Trapezoid
**Proof :** In $\Delta ABC$ and $\Delta CDE$, $AB=CD=c, BC= DE = a$ and
included $\angle ABC$ = included $\angle CDE$ \[As
$\angle ABC = \angle CDE = 90$°\]

Therefore, $\Delta ABC \cong \Delta CDR$. \[Side-Angle-Side theorem\]

∴$AC = CE = b$ and $\angle BAC = \angle ECD$.

Again, since $AB\perp BD$ and $ED\perp BD, AB \parallel ED$. Therefore,
$ABDE$ is a trapezoid.

$\angle ABC + \angle BAC + \angle ACB = 180$°

or, $90$°$+ \angle BAC + \angle ACB = 180$°

or, $\angle BAC + \angle ACB = 90$°

or, $\angle RCD + \angle ACB = 90$° \[As $\angle BAC = \angle ECD$\]

Again, $\angle BCD = 180$°

or, $\angle BCA + \angle ACE + \angle ECD = 180$°

or $\angle ACE = 90$°

∴$\Delta ACE$ is a right triangle.

Now, Trapezoid $ABDE = \Delta ABC + \Delta CDE + \Delta ACE$

or,
$\frac {1}{2} BD(AB+DE) = \frac {1}{2}(AB)(BC) + \frac {1}{2}(CD)(DE) + \frac {1}{2}(AC)(CE)$

or,
$\frac {1}{2} (a+c)(c+a) = \frac {1}{2}ac + \frac {1}{2}ac + \frac {1}{2}b^2$

or, $\frac {1}{2} (a+c)^2 = \frac {1}{2}(2ac + b^2)$

or, $(a+c)^2 = (2ac + b^2)$

or, $a^2 + 2ac + c^2 = 2ac + b^2$

or, $b^2 = a^2 + c^2$ ***\[Proved\]***

*\[Note : The 12th President of the United States of America, James A.
Garfield proved the Pythagorean Theorem with the help of two right
angled triangles. His proof of the Pythagorean Theorem was published on
page 161 of the New-England Journal of Education, April 1, 1876\][^2]*

### Proof with the help of similar triangles

### Proof with the help of Algebra

**Proposition :** Let, in the triangle $c$ is the hypotenuse and $a, b$
be the two other sides. It is required to prove that, $c^2=a^2+b^2$.
!Pythagorean Theorem\'s Proof with
Algebra
**Construction :** Draw four triangles congruent to $\Delta ABC$ as
shown in the figure.

**Proof :** The large shape in the figure is a square with an area of
$(a+b)^2$ \[As teh length of every side is same, $(a + b)$ and the
angles are right angles\]

The area of the small quadrilateral is $c^2$, as the shape is a square.
\[As, every side is $c$ and every angle of the quadrilateral is right
angle (proved while proving with the help of two right angled
triangles)\]

According to the figure, $(a+b)^2=4(\frac {1}{2}ab) + c^2$

Or, $a^2+2ab+b^2=2ab + c^2$

Or, $a^2+b^2=c^2$ ***\[Proved\]***

## Reference

[^1]: 1
    Byjus.com, Mathematics, Pythagoras Theorem Statement

[^2]: 2
    Sid J. Kolpas, \"Mathematical Treasure: James A. Garfield\'s Proof
    of the Pythagorean Theorem,\" Convergence (February 2016)


# Timeless Theorems of Mathematics/Law of Cosines

!Here
$c^2 = a^2+b^2-2ab\cdot \cos\gamma$

The law of cosines explains the relation between the sides and an angle
of a triangle. The law states that for any triangle $\Delta ABC$, if
$\angle ACB=\theta$, $AB=c$, $AC=b$ and $BC=a$, then
$c^2 = a^2+b^2-2ab\cdot \cos\theta$. For
$\theta=\frac{\pi}{2}, \Delta ABC$ is a right angle and
$c^2 = a^2 + b^2$, we have proved it as the pythagorean theorem earlier.

## Proof

### Statement

In any triangle, the square of one side\'s length is equal to the
difference between the sum of the squares of the other two sides\'
lengths and twice the product of those two sides\' lengths and the
cosine of the included angle.

Assume, $\Delta ABC$ is a triangle where $\angle ACB=\theta,$ $AB=c$
$AC=b$ and $BC=a.$ It is needed to be proved that,
$c^2 = a^2+b^2-2ab\cdot \cos\theta$.

Let us extend the line segment $BC$ to $BD$ (only for obtuse triangles),
where $AD\perp BD$. Assume $AD=h$ and $CD = k$.

### Proof with the help of the Pythagorean Theorem

#### For obtuse triangles:

!Construction for proving the law of cosines for angle
$\theta>\pi/2$.
According to the Pythagorean theorem, we can say that, for $\Delta ACD,$
$b^2 = k^2 + h^2$

Similarly, for the triangle $\Delta ABD,$ $c^2 = h^2 + (a+k)^2$
$= h^2 + a^2 + k^2 + 2ak$ $= a^2 + b^2 + 2ak$.

We will be using this value for further proof. But now, let\'s determine
some trigonometric values for the triangles.

Here, $\angle ACD = \pi - \theta$.

Therefore, $\frac{k}{b} = \cos(\pi -\theta)$

Or, $k = b\cdot\cos(\pi-\theta)$
$= b[\cos(\pi)\cdot cos(\theta) + \sin(\pi)\cdot \sin(\theta)]$
$= -b\cdot\cos(\theta)$

Now, $c^2 = a^2 + b^2 + 2ak$.

$\therefore c^2 = a^2 + b^2 - 2ab\cdot\cos\theta$. ***\[Proved\]***

#### For acute angles:

!Construction for proving the law of cosines for angle
$\theta<\pi/2$.
Like as the proof we have proved before, according to the Pythagorean
theorem, we can say that, for $\Delta ACD,$ $b^2 = k^2 + h^2$

Similarly, for the triangle $\Delta ABD,$ $c^2 = h^2 + (a-k)^2$
$= h^2 + a^2 + k^2 - 2ak$ $= a^2 + b^2 - 2ak$.

Here, $\angle ACD = \theta$.

Therefore, $\frac{k}{b} = \cos\theta$ $\Rightarrow k = b\cdot\cos\theta$

Now, $c^2 = a^2 + b^2 - 2ak$.

$\therefore c^2 = a^2 + b^2 - 2ab\cdot\cos\theta$. ***\[Proved\]***

*\[Note: Whatever the triangle is, the formula,
$c^2 = a^2 + b^2 - 2ab\cdot\cos\theta$ works.\]*


# Timeless Theorems of Mathematics/Brahmagupta Theorem

The Brahmagupta\'s theorem states that if a cyclic quadrilateral is
orthodiagonal (that is, has perpendicular diagonals), then the
perpendicular to a side from the point of intersection of the diagonals
always bisects the opposite side.[^1]

The theorem is named after the Indian mathematician Brahmagupta
(598-668).

## Proof

### Statement

If any cyclic quadrilateral has perpendicular diagonals, then the
perpendicular to a side from the point of intersection of the diagonals
always bisects the opposite side.

### Proof

!If $BM \perp AC$ and $EF\perp BC,$then $AF=FD,$ according to the
Brahmagupta\'s
theorem
**Proposition:** Let $ABCD$ is a quadrilateral inscribed in a circle
with perpendicular diagonals $AC$ and $BD$ intersecting at point $M$.
$ME$ is a perpendicular on the side $BC$ from the point $M$ and extended
$EM$ intersects the opposite side $AD$ at point $F$. It is to be proved
that $AF = DF$.

**Proof:** $\angle CBD = \angle CAD$ \[As both are inscribed angles that
intercept the same arc $CD$ of a circle\]

Or, $\angle CBM = \angle MAF$

Here, $\angle CMB + \angle CBM + \angle BCM = 180$°

Or, $\angle CMB + \angle BCM = 180$° $- \angle CBM$

Again, $\angle CME + \angle CEM + \angle ECM = 180$°

Or, $\angle CME + \angle CMB + \angle BCM = 180$° \[As
$\angle CMB = \angle CEM = 90$° and $\angle BCM = \angle ECM$\]

Or, $\angle CME + 180$° $- \angle CBM = 180$°

Or, $\angle CME = \angle CBM$

Or, $\angle AMF = \angle MAF$ \[As, \\angle AMF = \\angle CME; Vertical
Angles\]

Therefore, $AF = MF$

In the similar way, $\angle MDF = \angle DMF$ and $MF = DF$

Or, $AF = DF$ ***\[Proved\]***

## Reference

[^1]: Michael John Bradley (2006). The Birth of Mathematics: Ancient
    Times to 1300. Publisher Infobase Publishing. ISBN 0816054231. Page
    70, 85.


# Timeless Theorems of Mathematics/Napoleon's theorem

!$\Delta LMN$ is an equilateral
triangle
The Napoleon\'s theorem states that if equilateral triangles are
constructed on the sides of a triangle, either all outward or all
inward, the lines connecting the centers of those equilateral triangles
themselves form an equilateral triangle. That means, for a triangle
$\Delta ABC$, if three equilateral triangles are constructed on the
sides of the triangle, such as $\Delta ACM$, $\Delta BCX$ and
$\Delta ABZ$ either all outward or all inward, the three lines
connecting the centers of the three triangles, $ML$, $LN$ and $MN$
construct an equilateral triangle $LMN$.

## Proof

!A trigonomatric proof of the Napoleon\'s
theorem.
Let, $\Delta ABC$ a triangle. Here, three equilateral triangles are
constructed, $\Delta BCE$, $\Delta ABD$ and $\Delta ACF$ and the
centroids of the triangles are $P$, $Q$ and $R$ respectively. Here,
$AC=b$, $AB=c$, $BC=a$, $PQ=r$, $PR=q$ and $QR=p$. Therefore, the area
of the triangle $\Delta ABC$, $T=\frac{1}{2}bc\cdot\sin A$
$\Rightarrow bc\cdot\sin A=2T$

For our proof, we will be working with one equilateral triangle, as
three of the triangles are similar (equilateral). A median of
$\Delta ACF$ is $AG = (m+k)$, where $AR = m$ and $RG = k$. $AQ=n$ and,
as $\Delta ACF$ is a equilateral triangle, $\angle AGC=90^\circ$.

Here, $m+k=\frac{b\sqrt{3}}{2}$. As the centroid of a triangle divides a
median of the triangle as $1:2$ ratio, then
$m=\frac{2}{3}\cdot\frac{b\sqrt{3}}{2}$ $=\frac{b}{\sqrt{3}}$.
Similarly, $n=\frac{c}{\sqrt{3}}$.

According to the Law of Cosines, $a^2 = b^2 + c^2 - 2bc\cdot \cos A$
(for $\Delta ABC$) and for $\Delta AQR$,
$p^2 = m^2 + n^2 - 2mn\cdot\cos(A+60^\circ)$

$= (\frac{b}{\sqrt{3}})^2 + (\frac{c}{\sqrt{3}})^2 - 2\frac{b}{\sqrt{3}}\cdot\frac{c}{\sqrt{3}}\cdot\cos(A+60^\circ)$

$= \frac{b^2 + c^2 - 2bc\cdot\cos(A+60^\circ)}{3}$

$= \frac{b^2 + c^2 - 2bc(\cos A\cdot cos60^\circ - \sin A\cdot\sin 60^\circ)}{3}$

$= \frac{b^2 + c^2 - 2bc(\cos A\cdot \frac{1}{2} - \sin A\cdot\frac{\sqrt{3}}{2})}{3}$

$= \frac{b^2 + c^2 - bc(\cos A\cdot - \sin A\cdot\sqrt{3})}{3}$

$= \frac{b^2 + c^2 - bc\cos A\cdot - 2T\sqrt{3})}{3}$

$= \frac{a^2 + 2bc\cos A - bc\cos A - 2T\sqrt{3})}{3}$ \[According to
the law of cosines for $\Delta ABC$\]

$= \frac{a^2 + bc\cos A - 2T\sqrt{3})}{3}$

$= \frac{a^2 + \frac {b^2 + c^2 - a^2}{2} - 2T\sqrt{3})}{3}$

$= \frac{\frac {b^2 + c^2 + a^2 - 4T\sqrt{3}}{2}}{3}$

Therefore, $p = \sqrt{\frac{1}{6} (a^2 + b^2 + c^2 - 4T\sqrt{3})}$

In the same way, we can prove,
$q = \sqrt{\frac{1}{6} (a^2 + b^2 + c^2 - 4T\sqrt{3})}$ and
$r = \sqrt{\frac{1}{6} (a^2 + b^2 + c^2 - 4T\sqrt{3})}$. Thus,
$p = q = r$.

$\therefore\Delta PQR$ is an equilateral triangle. ***\[Proved\]***


# Introduction to Paleoanthropology/Definition

understand that it is a subdiscipline of
anthropology and have a basic understanding
of archaeology dating techniques, evolution of cultures, Darwinian
thought, genetics, and primate behaviours.

\_\_toc\_\_ There is a long academic tradition in modern anthropology
which is divided into four fields, as defined by Franz Boas (1858-1942),
who is generally considered the father of the Anthropology in the United
States of America. The study of anthropology falls into four main
fields:

1.  Sociocultural anthropology
2.  Linguistic anthropology
3.  Archaeology
4.  Physical anthropology

Although these disciplines are separate, they share common goals. All
forms of anthropology focus on the following:

-   Diversity of human cultures observed in past and present.
-   Human beings in all of their biological, social, and cultural
    complexity.
-   Many other disciplines in the social sciences, natural sciences, and
    even humanities are involved in study of human cultures.
    -   Examples include: Psychology, geography, sociology, political
        science, economics, biology, genetics, history and even art and
        literature.

## Sociocultural anthropology/ethnology

This field can trace its roots to processes of European colonization and
globalization, when European trade with other parts of the world and
eventual political control of overseas territories offered scholars
access to different cultures. Anthropology was the scientific discipline
that searches to understand human diversity, both culturally and
biologically. Originally anthropology focused on understanding groups of
people then considered \"primitive\" or \"simple\" whereas sociology
focused on modern urban societies in Europe and North America although
more recently cultural anthropology looks at all cultures around the
world, including those in developed countries. Over the years,
sociocultural anthropology has influenced other disciplines like urban
studies, gender studies, ethnic studies and has developed a number of
sub-disciplines like medical anthropology, political anthropology,
environmental anthropology, applied anthropology, psychological
anthropology, economic anthropology and others have developed.

## Linguistic anthropology

!Franz Boas is considered to be the founder of modern
anthropology.
This study of human speech and languages includes their structure,
origins and diversity. It focuses on comparison between contemporary
languages, identification of language families and past relationships
between human groups. It looks at:

-   Relationship between language and culture
-   Use of language in perception of various cultural and natural
    phenomena
-   Process of language acquisition, a phenomenon that is uniquely
    human, as well as the cognitive, cultural, and biological aspects
    involved in the process.
-   Through historical linguistics we can trace the migration trails of
    large groups of people (be it initiated by choice, by natural
    disasters, by social and political pressures). In reverse, we can
    trace movement and establish the impact of the political, social and
    physical pressures, by looking at where and when the changes in
    linguistic usage occurred.

## Archaeology

Is the study of past cultures through an analysis of artifacts, or
materials left behind gathered through excavation. This is in contrast
to history, which studies past cultures though an analysis of written
records left behind. Archaeology can thus examine the past of cultures
or social classes that had no written history. Historical archaeology
can be informed by historical information although the different methods
of gathering information mean that historians and archaeologists are
often asking and answering very different kinds of questions. It should
be noted that recovery and analysis of material remains is only one
window to view the reconstruction of past human societies, including
their economic systems, religious beliefs, and social and political
organization. Archaeological studies are based on:

-   A specific methodology to recover material remains including
    excavation techniques, stratigraphy, chronology, and dating
    techniques
-   Animal bones, plant remains, human bones, stone tools, pottery,
    structures (architecture, pits, hearths).

## Physical anthropology

Is the study of human biological variation within the framework of
evolution, with a strong emphasis on the interaction between biology and
culture. Physical anthropology has several subfields:

-   Paleoanthropology
-   Osteometry/osteology
-   Forensic anthropology
-   Primatology
-   Biological variation in living human populations
-   Bioarchaeology/paleopathology

### Paleoanthropology

As a subdiscipline of physical anthropology that focuses on the fossil
record of humans and non-human primates. This field relies on the
following:

Understanding Human Evolution:

Evolution of hominids from other primates starting around 8 million to 6
million years ago. This information is gained from fossil record of
primates, genetics analysis of humans and other surviving primate
species, and the history of changing climate and environments in which
these species evolved.

Importance of physical anthropology

Evidence of hominid activity between 8 and 2.5 million years ago usually
only consists of bone remains available for study. Because of this very
incomplete picture of the time period from the fossil record, various
aspects of physical anthropology (osteometry, functional anatomy,
evolutionary framework) are essential to explain evolution during these
first millions of years. Evolution during this time is considered as the
result of natural forces only.

Importance of related disciplines

Paleoanthropologists need to be well-versed in other scientific
disciplines and methods, including ecology, biology, anatomy, genetics,
and primatology. Through several million years of evolution, humans
eventually became a unique species. This process is similar to the
evolution of other animals that are adapted to specific environments or
\"ecological niches\". Animals adapted to niches usually play a
specialized part in their ecosystem and rely on a specialized diet.

Humans are different in many ways from other animals. Since 2.5 million
years ago, several breakthroughs have occurred in human evolution,
including dietary habits, technological aptitude, and economic
revolutions. Humans also showed signs of early migration to new
ecological niches and developed new subsistence activities based on new
stone tool technologies and the use of fire. Because of this, the
concept of an ecological niche does not always apply to humans any more.

## Summary

The following topics were covered:

-   Introduced field of physical anthropology;
-   Physical anthropology: study of human biology, non-human primates,
    and hominid fossil record;
-   Placed paleoanthropology within overall context of anthropological
    studies (along with cultural anthropology, linguistics, and
    archaeology);

Further modules in this series will focus on physical anthropology and
be oriented toward understanding of the natural and cultural factors
involved in the evolution of the first hominids.