OpenLLM-France/wikimedia huggingface dataset:
# Decimals

Decimals are basically fractions expressed without a denominator, rather replaced by a power of ten, and then the decimal point is inserted into the numerator at a position corresponding to the power of ten of the denominator. It is usual to add a leading zero to the left of the decimal point when the number is less than one.
    ${\frac {2}{5}}={\frac {2\times 2}{5\times 2}}={\frac {4}{10^{1}}}=0.4$

## Adding Decimal Numbers

Add decimal numbers much the same way you would add integers. Line up decimal points, and then proceed to add each column and carry at the top. The decimal point in the answer should line up with all of the others. Here is an example:
$12.3+24.2=36.5$
${\begin{aligned}{12.3}\\+{\underline {24.2}}\\{36.5}\end{aligned}}$

## Subtracting Decimal Numbers

Subtract as you would do with whole numbers, but remember to follow all the rules from addition of decimals.
$36.36-11.11=25.25$
    ${\begin{aligned}{36.36}\\-{\underline {11.11}}\\{25.25}\end{aligned}}$

## Converting Fractions to Decimal Numbers

To convert a fraction to a decimal number, divide the numerator by the denominator.
* ${\frac {3}{4}}=0.75$
* ${\frac {10}{3}}=3.333333333333333...$

## Repeating and Terminating Decimals

A repeating decimal is a decimal that is infinite. For instance, the 3.33... in the second example above. The threes just keep repeating.
Instead of writing many 3's, you can draw a line above the number that is repeating.
    ${\frac {10}{3}}={3.}{\overline {3}}$
When there are two numbers repeating, such as .232323, you have to draw a line above the 2 and the 3.
    ${0.}{\overline {23}}$
A terminating decimal is a decimal that ends at one point and does not go on forever. ex. 1.25

## Multiplying Decimal Numbers

Multiplying decimal numbers can be tricky at times, but most of the time, it is similar to multiplying any integers. Although there are easier methods of multiplying, this is one of the methods.
    You can make both decimal numbers have same multiple of a power of ten.
       $0.6\times 0.75=(60\times 10^{-2})\times (75\times 10^{-2})$
    Then multiply the first terms together, and the second terms.
       $(60\times 75)\times (10^{-2}\times 10^{-2})=4500\times (10^{-4})$
    Then insert the decimal point into a corresponding power of ten.
       $4500\times (10^{-4})=450\times (10^{-3})=45\times (10^{-2})=4.5\times (10^{-1})=.45\times (10^{0})=.45\times (1)=.45$

## Dividing Decimal Numbers

Dividing decimal numbers is similar to multiplying them.
    Make both decimal numbers have same multiple of a power of ten.
       $0.3/0.4=(3\times 10^{-1})/(4\times 10^{-1})$
    Then divide the first terms together, and the second terms.
       $(3\times 10^{-1})/(4\times 10^{-1})=(3/4)/10^{0}$
    Then insert the decimal point into a corresponding power of ten.
       $(3/4)/10^{0}=0.75/1=0.75\,\!$
Alternatively, you can make the numbers integers (if the decimal is finite) and perform a simple division.
       $0.3/0.4=(0.3\times 10)/(0.4\times 10)=3/4=0.75$
# Basic Physics of Digital Radiography
## Table of Contents

* Contents
* Introduction
* The Basics
    Atomic Structure || Electron Binding Energy || Electromagnetic Radiation || Diagnostic X-Rays || X-Ray Detection || Attenuation of X-Rays || Radiography Systems || The Fourier Transform
* The Source
    X-Ray Tube || X-Ray Energy Spectrum || HV Generator || Beam Filtration || Light Beam Diaphragm || Automatic Exposure Control
* The Patient
    Interaction Processes || Radiation Biology || Tissue Attenuation || X-Ray Shadows || Subject Contrast || Radiation Dose || Radiation Protection
* The Image Receptor
    Computed Radiography || Digital Radiography || XII-Video Systems || Radiation Dose Management
* The Computer
    Binary Representation || General-Purpose Computer || Digital Image Processor || Digital Images || Digital Image Processing || Image Storage & Distribution
* The Image
    Image Display || Image Contrast || Spatial Resolution || Image Noise || Detective Quantum Efficiency || Temporal Resolution || Image Quality Assessment
* The Applications
    General Radiography || Digital Subtraction Angiography || Cone-Beam Computed Tomography || Dual-Energy Radiography || Image Superimposition
* The Quiz
    X-ray Production & Interactions with Matter || Radiation Biology & Dosimetry || Technology of Radiographic Imaging
* Index

## Bibliography

* Bushberg JT, Seibert JA, Leidholdt & Boone JM, 2011. The Essential Physics of Medical Imaging, 3rd Ed..
* Gonzalez RC & Woods RE, 2008. Digital Image Processing, 3rd Ed..
* Heggie JCP, Liddell NA & Maher KP, 2001. Applied Imaging Technology, 4th Ed.
* Oppelt A, Ed., 2005. Imaging Systems for Medical Diagnostics.
# Basic Physics of Digital Radiography/Introduction

Digital Radiography heralds a new era for medical X-ray imaging. Radiographs can be recorded using digital image receptors and enhanced using computer processing. They can also be transferred to databases for archival and transmission throughout hospitals and clinics. The change from traditional film-based image receptors is similar in many respects to those occurring in digital photography and digital television. Greater precision can now be applied to each stage of the imaging process so that the balance between image quality and radiation dose can, at last, be controlled accurately. Knowledge and understanding of the basic physics which underlies practice is critical for its successful application in the clinical environment. This wikibook addresses this requirement.
The intent is a text which succinctly explains the physical basis of X-Rays and their modern application in Diagnostic Radiography. The wikibook is primarily for students with foundations in anatomy and physiology and could also be of interest to physics and engineering students requiring a topic overview. Math is kept to a minimum and only a basic knowledge of algebra is required of students. The use of calculus and other math techniques is largely avoided by using conceptual descriptions of the more complex processes. Keeping radiation dose as low as reasonably achievable while understanding its effects on image quality is the primary emphasis.
The focus extends from general radiography through fluoroscopy to 3D angiography as applied in general hospitals and clinics. The subject treatment is different from that used in conventional textbooks in that radiation biology is not treated as a separate topic but is embedded within the consideration of image formation. Likewise, the topic of image quality is integrated into the treatment of image display. The hope is that more integrated treatment of the subject is developed for those studying the field for the first time.
Note that the particular requirements of mammography and of paediatric, dental, chiropractic and veterinary radiography are not covered at this time. A companion WikiBook on the Basic Physics of Nuclear Medicine is also available.
# Glossary

Conversion
Greatest Common Factor
Improper Fraction
Least Common Multiple
Liter - The basic unit of volume in the metric system
Multiplication
Meter - The basic unit of length in the metric system.
# Measurements
## Converting English Measurements

Here are some conversions that you should know in the English system:

### Length

    1 foot = 12 inches 1 yard = 3 feet 1 mile = 5280 feet 1 gallon = 16 cups 1 gallon = 4 quarts

### Mass

1 pound = 16 ounces
1 ton (in United States) = 2000 pounds

## The Metric System

Metric system is based on the SI (Système International d'Unités) units.

### Length

    Length is measured in meters (m). There are 100 centimeters (cm) in 1 meter, and 10 millimeters (mm) in each centimeter. Therefore, there are 1000 millimeters in each meter. There are 1000 meters in each kilometer (km).

### Mass

    Mass is measured in kilograms (kg). The kilogram was originally defined as the mass of one liter of pure water at a temperature of 4 degrees Celsius and standard atmospheric pressure.

### Time

    Time is measured in seconds (s). Second is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium-133 atom at zero kelvins. There are 60 seconds in a minute, 3600 seconds in an hour (or 60 minutes), and 86,400 seconds in a day (or 24 hours, or 1,440 minutes). There are 1000 milliseconds in a second.

### Temperature

    Temperature is measured in Kelvins or degrees Celsius (°C) in SI; the latter is used more frequently in general applications. The degree Celsius is a unit of temperature named after the Swedish astronomer Anders Celsius (1701–1744), who first proposed a similar system in 1742. The Celsius temperature scale was designed so that the freezing point of water is 0 degrees Celsius, and the boiling point is 100 degrees Celsius at standard atmospheric pressure.

### Volume

    Volume is measured cubic meter (m$^3$). The volume of a solid object is a numerical value given to describe the three-dimensional concept of how much space it occupies. One-dimensional objects (such as lines) and two-dimensional objects (such as squares) are assigned zero volume in the three-dimensional space. Liter (L and l) is used commonly. 1000 L makes a cubic meter and 1 L is 1 dm³.

## Metric Length Measurement Conversions

1 meter = 10 decimeters = 100 centimeters = 1000 millimeters 1 decimeter = 10 centimeters = 100 millimeters 1 centimeter = 10 millimeters 1 kilometer = 1000 meters = 10,000 decimeters = 100,000 centimeters = 1,000,000 millimeters 1 kilogram = 1000 grams = 1,000,000 milligrams 1 gram = 1000 milligrams

## Converting English and Metric Measurements

To convert between one measurement unit and another, requires a conversion factor. For instance;
1 inch = 2.54 cm or 25.4mm
and so 12 inches = 12 x 25.4 = 304.8mm
The same conversion factor can also be used to convert mm to inches by using division. For example;
400mm = 400/25.4 = 15.75 inches (to 2 decimal places)
# Basic Math for Adults

Welcome to the Basic Math For Adults wikibook. This book covers basic arithmetic concepts including whole numbers, decimals, fractions, and English and metric measurements. Instructional emphasis is on application to typical problems one would encounter in the workplace. Simple and yet effective algorithms (methods) are used to solve them. Furthermore, you should notice that all digits here involve the base 10 numbering system.

## Chapters

* Basic Terminology (Includes decimals and numerals)
* Basic Whole Numbers (Rounding, Adding, Subtracting, Multiplying, Dividing and Factoring Whole Numbers)
* Fractions
* Decimals
* Measurements
* Glossary
# Basic Math Terminology
## The Decimal System

The decimal system, sometimes referred to as base 10, contains a total of ten identifiers called digits. The decimal system is widely used because humans generally have ten fingers on which to count. For now, we will disregard other number systems for the sake of simplicity with the understanding that the decimal system is not unique.
The ten digits of the decimal system, arranged from lowest to greatest, are:
* 0 (zero)
* 1 (one)
* 2 (two)
* 3 (three)
* 4 (four)
* 5 (five)
* 6 (six)
* 7 (seven)
* 8 (eight)
* 9 (nine)
The decimal system uses positional notation to represent numbers larger than 9. This means that a digit's position in relation to other digits affects its meaning. Digits in the furthest right position represent the number of ones being counted, while digits in the second position from the right represent the number of tens. Digits in the third position from the right represent the number of hundreds, and digits in the fourth position from the right represent the number of thousands. This pattern can continue forever; for more information, see orders of magnitude.
For example, the number 535,254 means 5 hundreds of thousands, 3 tens of thousands, 5 thousands, 2 hundreds, 5 tens, and 4 ones. We would say this number as "five hundred thirty-five thousand two hundred fifty-four".
Other systems, like binary (base 2) exist. Base 2 would just have 2 digits, 0 and 1, base 3 has 3 digits, 0 ,1 and 2, and so on.

## The Basic Sets of Numerals

Counting numbers are the numbers we use every day to count things. Mathematicians sometimes refer to this set of numbers as the Natural Numbers. These are represented by the sign $\mathbb {N}$, for Natural Numbers.
    $1,2,3,4...\$
Whole numbers include all the counting numbers and zero.
    $0,1,2,3,4...\$
Negative numbers include the opposite of all the Counting Numbers. They are counted in the opposite direction of Counting numbers and have -, the negative sign, in front of them.
    $...-4,-3,-2,-1$
Integers include all numbers without a decimal. Another way to say this is all of the Whole Numbers and their negatives. These are represented by the sign $\mathbb {Z}$, for the German word Zahlen, which means "numbers".
    $...-4,-3,-2,-1,0,1,2,3,4...$
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6/6. Nxf6

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6...Qxf6 is a useful developing move - many writers caution against bringing the queen out early, but without the b1-knight on the board she doesn't have to worry about losing tempos to Nb1-c3-d5 or Nb1-c3-e4 ideas. Any opening that features a quick ...e5 and ...f5 (looking at you, Latvian Gambit) will likely involve putting the queen on f6 and castling kingside to bring a rook to f8 and then mating White on the f-file.
6...gxf6 has been played by FM Danko Bokan. You get the same pawn structure as in the 6.Qe2 d5 7.Nxf6+ line but White still has the option of putting her queen on a square other than e2.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6/6. Nxf6/6...Qxf6/7. Qe2/7...Be7

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White accepts the gambit with 8. Bxc6, planning to capture on e5 once the guard is removed.
Castling kingside is still sub-optimal due to the ...Nd4 option (compare the position after 6...Qxf6).
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6/6. Nxf6/6...Qxf6/7. Qe2/7...Be7/8. Bxc6/8...bxc6/9. Nxe5

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* 9...O-O looks like the best developing move Black has, immediately bringing the rook to the open f-file. What could be better than that?
* 9...Qe6 could be better than that. Catching the opposing king in the centre is half the fun of sacrificing pawns, and here 10.O-O?? would unprotect the White queen and so drop a piece to 10...d6.
** 10.d4 is surprisingly popular given that it doesn't actually get out of the pin! (It does help develop a piece though.) Black has tried various moves to claim some initiative; Radjabov's choice is 10...Bf6, 10...O-O is also obvious, 10...a5 prepares ...Ba6 hitting the queen, 10...c5 immediately challenges the d-pawn.
** Karjakin tried 10.Nf3 against Radjabov, which does get out of the pin. White loses castling privileges after 10...Qxe2 11.Kxe2 but can easily castle by hand with Re1 and Kf1.
** Dominguez Perez tried 10.Nc4 against Fressinet, to similar effect.
** 10.Qe3 is a radical solution tried by Frolyanov and Timofeev. Black has ...Ba6, ...Bf6 and ...c5 options after mutual kingside castling.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6/6. Nxf6/6...Qxf6/7. Qe2/7...Be7/8. Bxc6

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Here's an example of how chess opening theory moves with the times.
* 8...dxc6 was by far the more common move until the mid 2000s. It seems logical to open a diagonal for the bishop that it can actually use, right? The same exchange in the Exchange Ruy Lopez is always met with dxc6.
* But then along came Teimour Radjabov and revived the whole 5...Nf6 line based on the innovation 8...bxc6, and now every top player who tries this line - Zvjagintsev, Timmerman, Fressinet - goes 8...bxc6.
Why 8...bxc6 and not 8...dxc6? The intelligent answer is something to do with capturing towards the centre and pawn islands. The honest answer is that Black players tried 8...dxc6 and laid an egg, so they tried something different.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6/6. Nxf6/6...Qxf6/7. Qe2/7...Be7/8. Bxc6/8...bxc6

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* 9. Nxe5 is the main line. White simply captures the free pawn with the nearest piece to hand.
* 9. Qxe5 has been played by Michael Adams, because being a pawn up and taking the queens off is usually what you want to do, but in this case simplifying hurts White's winning percentage as well as Black's. The queen enjoys being the most mobile piece on the board; if you exchange her off then that title would pass to the Black bishops, and there are two of them.
* 9. d4 is a fun idea of Swedish GM Emanuel Berg's, taken up by Alexei Shirov. It's not a gambit in that if Black grabs the pawn with 9...exd4, White shoves the queen away with 10.Bg5 and grabs it back. It's a pseudo-gambit in that White had the chance of going 1 pawn up and instead decided to be 0 pawns up.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6/6. Nxf6/6...Qxf6/7. Qe2/7...Be7/8. Bxc6/8...dxc6

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9. Nxe5 is the main line. White needs to capture the pawn now otherwise Black will play ...Bd6 and ...Bg4 and the window will be shut.
Why is it not a good idea to capture with 9. Qxe5, since White will be a pawn up with the queens off? Because Black doesn't have to obligingly exchange queens on e5. Instead she can play 9...Bg4 to exploit the brittle knight on f3, which won't have any good squares after 10.Qxf6 Bxf6, and if White has to play gxf3 her kingside pawn majority will be nobbled. (10.Qxc7? is suicide with White so under-developed. The simple 10...O-O! gives Black an irresistible attack.)
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6/6. Nxf6/6...Qxf6/7. Qe2/7...Be7/8. Bxc6/8...dxc6/9. Nxe5

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9...Bf5 is a useful developing move. 9...O-O is also a useful developing move.
Why is ...Bf5 more popular than ...O-O? It comes down to a general principle of opening theory: postpone the hard decisions for as long as you can. Black in this position no longer has to decide where to put the c8-bishop. The White knight is hitting d7 and g4, and on e6 the bishop will be vulnerable to tactics like Nxc6 as soon as White gets a rook to the e-file. So it has to go to f5. That's an easy decision. Whether Black wants to castle queenside or kingside is a hard decision. Make the easy decision first!
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6/6. Nxf6/6...Qxf6

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Having avoided committing the queen to e2 on move 6, White typically puts it there anyway with 7. Qe2. This restricts Black's options, because now ...d5 would self-pin the c6-knight allowing White to capture the e5-pawn for zero compensation.
Viktor Kupreichik mixed things up a couple times with 7. O-O. If Black reacts the same way she would after 7. Qe2, with 7...Be7, White has a huge positive score with the plan of Bxc6 and Qe1. But alert Black players found 7...Nd4! to take advantage of the fact that the e-pawn isn't under any pressure.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6/6. Nxf6/6...Qxf6/7. Qe2

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Attempting to defend the e-pawn with moves like 7...d6 or 7...Bd6 are ultimately futile. Either 7...d6 8.d4 or 7...Bd6 8.d4 (not allowing Black the trick 8.Bxc6 dxc6 9.d4 Bg4) just win it outright.
So instead Black gambits the pawn with 7...Be7. The compensation will be in the form of open lines for the rooks and an unopposed light-squared bishop.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...Nf6

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Black has challenged White's unprotected e4-knight with another knight. This is a problem and an opportunity for White. In the 5...d5 lines this knight doesn't have the opportunity of gaining a tempo by swapping itself for a Black piece of equal value. Here, it does! So an obvious move here would be
Why might White have second thoughts about 6.Nxf6+? After all, if White defends the knight then it's 100% going to get kicked away from e4 on the very next move with ...d5. The answer may lie in Black's options for recapturing on f6. For reasons of both development and pawn structure it looks as though Black would prefer to use the queen and not the g-pawn, but if White now indirectly threatens Black's e-pawn with
then 6...d5 7.Nxf6+ Qxf6? doesn't work because 8.Qxe5+ would just pick up a pawn, so Black would have to go 7...gxf6 instead. The immediate 6.Nxf6+ could be met effectively by 6...Qxf6.
This is excellent logic, but chess opening theory doesn't always reward logic. The more honest answer to whether 6.Qe2 is better than 6.Nxf6+ is "try them and see!" 6.Nxf6+, for one reason or another, outscores 6.Qe2 both in White wins and Black defeats.