Number Systems and Bases – BetterExplained Home › Programming ›Number Systems and Bases Base systems like binary and hexadecimal seem a bit strange at first. The key is understanding how different systems “tick over” like an odometer when they are full. Base 10, our decimal system, “ticks over” when it gets 10 items, creating a new digit. We wait 60 seconds before “ticking over” to a new minute. Hex and binary are similar, but tick over every 16 and 2 items, respectively. Try converting numbers to hex and binary here: Way back when: Unary Numbers Way back in the day, we didn’t have base systems! It was uphill both ways, through the snow and blazing heat. When you wanted to count one, you’d write: l When you wanted 5, you’d write lllll And clearly, 1 + 5 = 6 l + lllll = llllll This is the simplest way of counting. Enter the Romans In Roman numerals, two was one, twice. Three was one, thrice: one = I two = II three = III However, they decided they could do better than the old tradition of lines in the sand.

For five, we could use V to represent lllll and get something like l + V = Vl Not bad, eh? And of course, there are many more symbols (L, C, M, etc.) you can use. The key point is that V and lllll are two ways of encoding the number 5. Give each number a name Another breakthrough was realizing that each number can be its own distinct concept. Rather than represent three as a series of ones, give it its own symbol: “3″. Do this from one to nine, and you get the symbols: 1 2 3 4 5 6 7 8 9 The Romans were close, so close, but only gave unique symbols to 5, 10, 50, 100, 1000, etc. Use your position Now clearly, you can’t give every number its own symbol. There’s simply too many But notice one insight about Roman numerals: they use position of symbols to indicate meaning. IV means “subtract 1 from 5″ and VI means “add 1 to 5″. In our number system, we use position in a similar way.

We always add and never subtract. And each position is 10 more than the one before it. So, 35 means “add 3*10 to 5*1″ and 456 means 4*100 + 5*10 + 6*1. Our choice of base 10 Why did we choose to multiply by 10 each time? Most likely because we have 10 fingers. One point to realize is you need enough digits to “fill up” until you hit the next number. Let me demonstrate. If we want to roll the odometer over every 10, so to speak, we need symbols for numbers one through nine; we haven’t reached ten yet. Imagine numbers as ticking slowly upward – at what point do you flip over the next unit and start from nothing? Enter zero And what happens when we reach ten? How do we show we want exactly one “ten” and nothing in the “ones” column? We use zero, the number that doesn’t exist. Zero is quite a concept, it’s a placeholder, a blank, a space, and a whole lot more.

Suffice it to say, Zero is one of the great inventions of all time. Zero allows us to have an empty placeholder, something the Romans didn’t have. Look how unwieldly their numbers are without it. George Orwell’s famous novel “1984″ would be “MCMLXXXIV”! Rolls right off the tongue, doesn’t it? Considering other bases Remember that we chose to roll over our odometer every ten. Our counting looks like this: 1 2 3 4 5 6 7 8 9 (uh oh, I’m getting full!) 10 (ticked over – start a new digit) What if we ticked over at 60 when we counted, like we do for seconds and minutes? 1 second 2 3 4 5 … 58 59 1:00 (60 seconds aka 1 minute. We’ve started a new digit.) Everything OK so far, right? Note that we use the colon (:) indicate that we are at a new “digit”. In base 10, each digit can stand on its own.

Try Base 16 If we want base 16, we could do something similar: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 (we’re getting full) 1:00 (16 – we’ve started a new digit) However, we don’t want to write hexadecimal numbers with the colon notation (even though we could). We’d rather cook up separate symbols for 10-15 so we can just write numbers like we’re used to. We’ve run out of numbers (1-9 already used, with 0 as a placeholder) so we need some other symbols. We could use some squiggly lines or other shapes, but the convenions is to use letters, Roman style. Just like 5 became V, programmers use letters A-F to get enough digits up to 16.

That is, 1 2 3 4 5 6 7 8 9 A (10 – we’re using the symbol “A”) B (11) C (12) D (13) E (14) F (15 – uh oh, we’re getting full) 10 (16 – we start a new digit) Ahah! Now we can use one digit per “place”, and we know that 10 actually means we’ve “ticked over to 16″ once. 20 means we’ve ticked over to 16 twice (32). 25 means we’ve ticked over to 16 twice (giving us 32) and gone an extra 5. The total is 32 + 5 = 37. Quick review With me so far? This is pretty cool, right? We can count in any system we want. Also notice that base 16 is more “space efficient” in the sense we can write a number like 11 in a single digit: B. Base 16 really isn’t that different from base 10, we just take longer to fill up.

The wonderful world of binary We’ve seen plenty of base systems, from over-simple unary, to the unwiedly Roman numerals, the steady-going base 10 and the compact base 16. What’s great about binary? In the spirit of keeping things simple, it’s the simplest number system that has the concept of “ticking over”. Unary, where we just write 1, 11, 111… just goes on forever. Binary, with two options (1 and 0) looks like this: 1: 1 2: 10 (we’re full – tick over) 3: 11 4: 100 (we’re full again – tick over) 5: 101 6: 110 7: 111 8: 1000 (tick over again) … and so on. Because binary is so simple, it’s very easy to build in hardware. You just need things that can turn on or off (representing 1 and 0), rather than things that have 10 possible states (to represent decimal). Because it’s so simple, binary is also resistant to errors.

If your signal is “partially on” (let’s say 0.4), you can assume that’s a zero. And if it’s mostly on (say 0.8), then you can assume it’s a 1. If you’re using a system with 10 possible states, it’s difficult to tell when an error occurred. This is one reason digital signals are so resilient to noise. Other examples of bases We use other bases all the time, even dynamically changing bases. We usually don’t think of it that way: Hours, minutes, seconds: 1:32:04 We know this is 1 hour, 32 minutes, 4 seconds. In seconds, this is 16060 + 32*60 + 4. Feet and inches: 3′ 5″ This is 3 feet, 5 in or 3 * 12 + 5 inches. Pounds and ounces: 8 lbs, 5 oz Since a pound is 16 oz, This is 8 * 16 + 5 oz. We’ve been using a base 16 number system all along!

Parting thoughts “10″ in any number system indicates the base, and means we’ve ticked over once. 10 in binary means two, 10 in decimal means ten, and 10 in hexadecimal is sixteen. How do you keep these numbers apart? Programmers will often write “0b” in front of binary numbers. So 2 in binary is 0b10 Similarly, they’ll write 0x in front of hex numbers. So 16 in hex is: 0×10 If there aren’t any symbols (0b or 0x) in front, we assume it’s base 10, a regular number. Now go forth and enjoy your new knowledge! Base 8 - the best number system. So long, Base 10! Back to index of science pages by Donald Sauter. Base 8 - the Best Number System! Right off the bat let me pull the rug out from under you and freely admit myself that this idea probably won't be implemented tomorrow. Looks like Base 10 kind of has a toehold. Still, that shouldn't hold us back from the fun of thinking and comparing and analyzing.

Nothing's impossible, and the adoption of Base 8 some day by some civilization is quite a few steps up from unthinkable. Suppose, for example, some four-fingered aliens or, more horrifying yet, another French Revolution, came along and forced it on us, along with 10-day weeks and 1000-day years, shudder, shudder. (Well, at least the former scenario can be proven to be impossible - see my "Human Race is Special" page.) We humans are "binary" creatures. By that I mean that we can only reliably double and halve quantities using our senses. To be honest, I'm really only referring to lengths here, but that's one of our most common and fundamental measurements. If you have a stick, or a board, or the edge of a table, or a wall, or a column of water, etc., you can reliably put your finger on the midpoint. You'll know you have it right because if you didn't there would be a visible imbalance between the two sides.

So far, that doesn't sound like a big deal; you can hit the midpoint whether you're living in a Base 10 world, where the halfway mark is called .5, or a Base 8 world, in which it's called .4 (because 4 is half of 8.) But in Base 8 you can continue halving halves to zoom in on any number that can be written. For example, suppose you needed to mark the .31 spot on this unit length: ________________________________________________________________ We concentrate first on locating .3 . The halfway point is .4 : _______________________________|________________________________ .4 We find .2 halfway between 0 and .4 : _______________|_______________.________________________________ .2 And .3 is halfway between .2 and .4 : _______________._______|_______.________________________________ .3 Now we go for the second digit.

We split .3 and .4 to get .34 : _______________._______.___|___.________________________________ .34 We find .32 halfway between .3 and .34 : _______________._______._|_.___.________________________________ .32 And, finally, we nail .31 halfway between .3 and .32 : _______________._______.|._.___.________________________________ .31 Imagine that, two-place precision without any sort of measuring device! And with the point of your pen, you could take it to another place, such as .316. If you haven't thought about this before, you should be pretty impressed. And remember, what we just did could be done for any number expressed octally. Even if we were given a number with many digits after the decimal point (I know I'm using a lot of Base 10 terminology here) we would very quickly zoom in on the spot where our fingertip doesn't move anymore. On the other hand, what we just did cannot generally be done in Base 10. That requires guesswork.

You can only take a stab at the .1, .2, .3, .4, .6, .7, .8, or .9 spots. Yes, you can nail the numbers that occur naturally in Base 8, such as 1/2 (.5), and multiples of 1/4 (.25), 1/8 (.125), etc., but just those few. Think about the times when you've needed to find the .7 spot between the smallest divisions of some scale. I'll bet what you do is two quick halvings to get to .75 and then say, "Ok, it's a little less than that." If you needed .8, you would move up "a little" from .75 . (By a similar process of doubling we could quickly construct any desired length greater than 1 using the unit length. But I don't think this has as much practical significance, and notice that what we would do in Base 10 borrows directly from the binary method, anyhow, always being mindful of the powers of 2.

For instance, if you needed to construct a length of 19 from a unit, you would do four doublings to get to 16, add on a single doubling to get 18, and then add on one unit. In Base 8, the representation for 19 is 23, which shows directly that we need 2 x 8, plus 3 units.) In third grade the teacher had us folding paper into fractions. Getting exact halves was a cinch, but when Mrs. Koehler challenged us to fold a piece of paper in thirds, I got very frustrated. In Base 8, 1/3 is .252525..., and by continual halvings you can zero in on that. (I just did an experiment - with excellent results. So there.) The point of this is, and I wish I could conjure up the most compelling words, is that in Base 8 we can connect every number to the real world just using our guts. This is only true for number systems based on powers of 2, so it would also be true for number systems based on 2, 4, and 16.

I don't think I'd get much of an argument by saying that Base 8 represents the best compromise between the purity of Base 2 and the compactness and convenience of Base 10. In fact, moving from Base 10 to Base 8 would hardly give rise to the need for more digits. Two octal digits are almost as precise as two decimal digits. I suppose someone could put my feet to the fire and argue that our senses are capable of dividing into thirds as well. But that would take both hands to mark the third-way points, require much more time and effort to gauge the equalness of the three segments, and yield correspondingly less reliable results. (I just tried it on a piece of paper and missed by a mile. So there.) Notice how builders naturally went for the binary system of halving halves. An inch is divided into halves, quarters, eighths, sixteenths, thirty-secondths, and sixty-fourths. A superb idea but, with all due respect, the fractional notation is strictly for the birds. All those numbers would look great in Base 8.

A really messy one like 37/64, which is .578125 in decimal notation, works out to a slim and trim .45 in octal notation. A nice side benefit of Base 8 is that the multiplication table is 64% as big as the Base 10 multiplication table. The same is true of the addition table. So arithmetic would be that much easier for everybody. In his book "The Realm of Numbers" Isaac Asimov spends some time extolling the virtues of Base 12: Observe how useful the dozen is. If you have a dozen apples, you can divide them equally into 2 groups of 6 each, 3 groups of 4 each, 4 groups of 3 each, 6 groups of 2 each, or 12 groups of 1 each. The important thing is that not only are 2 and 4 [as in number systems based on 16] factors of 12, but 3 is also. . . . The convenience of such factoring in practical arithmetic is great, and there are people who wish that we had used 12 as the base of our number system instead of 10.

The number 10 has only two factors, 2 and 5, and cannot be divided evenly by either 3 or 4. The only reason 10 won out over 12 is probably the anatomical accident that we have 5 fingers on each hand. Now if we had had 6 on each hand -- I say the importance of a factor of 3 is grossly trumped up; that it is not something we have a natural feel for, and division by 4, which is very natural in Base 8, gives something very similar anyhow. If a factor of 3 is so important, why not 5 and 7, etc? Moreover, Base 12 all of a sudden gives us multiplication and addition tables 44% larger than in Base 10 - yeowww! I say, now if we had only ignored our thumbs -- As admitted at the beginning, we're not likely to convert over tomorrow with a big smile on our face. And it certainly wouldn't be worth the effort if we're going to wipe ourselves out or revert to cavemen in the next few decades. It's hard to envision any given generation saying, "Sure, we'll take a dive for future generations.

It's for all the kids, man." I'm guessing it'll be more than just the grannies wondering what happened to all the numbers between 77 and 100. But to pull this out of the depths of complete impossibility, I offer a plan. The plan is simply a new set of numerals for Base 8. In that way, Base 10 and Base 8 could coexist until the time comes when Base 10 has faded silently away forever (or until 3-fingered aliens descend upon us.) After all, we seem to survive with lots of different units of measure going on. Do we ever. (See my proposal for fixing the units mess.) What occurred to me was to flip our current numerals left-for-right.

It works out nice that 2 through 7 are not symmetric about a vertical axis, and 8, which is symmetric, is not used in Base 8 (which will be called Base 10, heehee.) 0 and 1, in their basic forms are symmetric, but they need new characters, anyhow, as they do in Base 10, to finally eliminate the mess caused by the identity with alphabetic characters - ohs and zeros and ells and ones and eyes, oh my. To refresh your memory, here are the numerals from 0 to 7 (in reverse order - why did I do that, I wonder?): Here they are flipped left-for-right. (I didn't say mirror image. Mirrors don't flip left and right, but we won't get into that here. Ok, we will, since I brought it up. It's us that flip left and right when we turn something toward a mirror.

A mirrors pulls the object through itself, back to front, which has the effect of making a right-handed object look left-handed.) And tweaking them a bit for the sake of writeability and uniqueness, this is what I ended up with for our Base 8 numerals: Just an idea; this doesn't have to be the final word. I suppose now you want me to tell you how to say the Base 8 numbers, huh? To be honest, I hadn't given it any thought for all these years until I got to this point in this web page. But oot answers come right to mind, the nuwhth being ever so slightly tongue in cheek - we can pronounce the numbers backwards, too. That's not hard: oh-reez', nuwh, oot, eethr, orf, vee-ahf', skiss, and nev-ess'. How's that for English chauvinism? The ootth answer is to leave this to the people putting together our universal second language (see my page on that) , since Base 8 will be the official number system in that language, right?

Then the problem is solved once and for all for everybody, as opposed to new and different words being added to each of the thousands of languages out there. If the panel wants a bit of advice, I would suggest that there be only pronunciations for numbers, no associated written words. In the universal second language numbers will only be written as numerals. Numerals will be considered elemental, as letters are for words, and representing numerals with letter combinations will be viewed as nonsensical.

So you can hit the ground running when Base 8 comes along, here is the addition table: Base 8 Addition Table 0 1 2 3 4 5 6 7 10 1 2 3 4 5 6 7 10 11 2 3 4 5 6 7 10 11 12 3 4 5 6 7 10 11 12 13 4 5 6 7 10 11 12 13 14 5 6 7 10 11 12 13 14 15 6 7 10 11 12 13 14 15 16 7 10 11 12 13 14 15 16 17 10 11 12 13 14 15 16 17 20 Got it? 7 + 5 = 14. Now here's the multiplication table.

WARNING: Figuring that everybody would give this one glance and surf off into the wild blue yonder, I planted an error or so to snag a few visitors into taking a closer look. Base 8 Multiplication Table 1 2 3 4 5 6 7 10 2 4 6 8 12 14 16 20 3 6 11 14 17 22 25 30 4 10 14 20 24 30 34 40 5 12 17 24 31 36 43 50 6 14 22 30 36 44 52 60 7 16 25 34 43 52 63 70 10 20 30 40 50 60 70 100 5 x 5 = 31 . . . what a gas!

Notice it's multiplication by 4 that gives the nice sequence of answers alternately ending in 0 and 4. *** DON'T GET LEFT BEHIND! You can get started in Base 8 by leaving out all the 8's and 9's while trying to fall to sleep tonight: 1 sheep . . . 2 sheep . . . 3 sheep . . . 4 sheep . . . 5 sheep . . . 6 sheep . . . 7 sheep . . . 10 sheep . . . 11 sheep . . . 12 sheep . . . 13 sheep . . . 14 sheep . . . 15 sheep . . . 16 sheep . . . 17 sheep . . . 20 sheep . . . 21 sheep . . . 22 sheep . . . 23 sheep . . . 24 sheep . . . 25 sheep . . . 26 sheep . . . 27 sheep . . . 30 sheep . . . 31 sheep . . . 32 sheep . . . 33 sheep . . . 34 sheep . . . 35 sheep . . . 36 sheep . . . 37 sheep . . . 40 sheep . . . 41 sheep . . .

42 sheep . . . 43 sheep . . . 44 sheep . . . 45 sheep . . . 46 sheep . . . 47 sheep . . . 50 sheep . . . 51 sheep . . . 52 sheep . . . 53 sheep . . . 54 sheep . . . 55 sheep . . . 56 sheep . . . 57 sjeep . . . 60 sheep . . . 61 sheep . . . 62 sheep . . . 63 skeep . . . 64 sheep . . . 65 sheep . . . 66 sheep . . . 67 sseep . . . 70 sheep . . . 71 sweep . . . 72 sneep . . . 73 sheep . . . 74 smeep . . . 75 sheep . . . 76 sheep . . . 77 sbeep . . . 100 sheep . . . 101 szeep . . . szzzzzzz . . . Number Systems Different-Base Number Systems This page will explain the significance of different-base number systems, and how important they are to the field of computing.

Introduction The base number system we are all familiar with is 10 [Decimal]. With it we use ten unique integers: 0 1 2 3 4 5 6 7 8 9 After the number nine comes 10. Notice that 10 is not a unique number; it is a combination of two unique numbers. The numbers 10-19 are the second iteration of the base 10 system, 20-29 are the third iteration, and so forth. The base 8 system (Octal) uses only the integers 0-7, so counting with it would look like this: 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 ... After 7 comes 10, which starts the second iteration in the octal system. 10 in octal is not the same as 5+5; you can see it is the equivalent of 8 in decimal. Likewise, 20 octal is 16 in decimal, 100 octal is 64 decimal.

Now we take a look at the base 2 number system [binary]. The only unique integers in this system are 0 and 1. Counting with it looks like this: 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 ... The number 10000 binary is the equivalent of 16 in decimal and 20 in octal. We have looked at the decimal, octal and binary number systems. When a base-number system is greater than 10, additional integers besides 0-9 must be used. The base 16, or hexadecimal system, is a good example. Because we don't have 16 unique integers in our numbering, we use letters to fill integers after 9.

So hexadecimal counting looks like this: 0 1 2 3 4 5 6 7 8 9 A B C D E F 10 11 12 13 14 15 16 17 18 19 1A ... E4 E5 ... F9 FA ... A look at different base systems: When expressing large numbers, the binary system is very inefficient, as is any low base system. Higher base systems can express numbers using fewer characters. To illustrate this, we will look at a table showing numbers as they would look in different base systems. The columns are different systems: Base 2, 3, 4, 8, 9, 10, 16, and 18. The rows are different numbers as they look in each system. *2* 1A8 Note: The commas are present only to simplify reading Notice that it is much simpler to express numbers using larger base systems. Because computers rely on the binary system, base 16 is used for expressing long binary numbers. There is a reason base 16 is used for binary instead of say 10, or 18.

If you look at the table, 100,000,000 binary is the same as 100 hexadecimal, whereas it is '256' in base 10 and 'E4' in base 18. Hexadecimal is very compatible with binary because 16 is a power of 2. Likewise, base 4 and 8 are also very nice systems to represent binary numbers. This is not only applicable with powers of 2 though. All of the base 3 numbers in the table work well with base 9 (though the only number showing this obviously in the table is '1000'). One thing to note is that base 10 is not "nice" with respect to base 2, and base 18 is not "nice" with respect to base 3, though each is a multiple of the previous. In order for number systems to "look good together", the higher system must be a power of the lower. What does this all mean? Understanding different number systems is essential to understanding how computers work. A standard personal computer contains millions of transistors, each of which either contains or doesn't contain an electrical charge.

This property of transistors makes them perfect containers for the base 2 system--0 and 1 (symbolic of "no charge" and "electrically charged," also known as "off" and "on"). When millions upon millions of these building blocks are used to create a computer system, it becomes a complex machine which is and handles binary data. And because of this, everything a computer does can be simplified to binary operations. Example: When you add 3 and 5 on a computer, the computer is really adding what is contained in a total of five transistors: ("1" "1") for the number 3, and ("1" "0" "1") for the number 5. The trick of "lining up" transistors like this allows us to use them to represent binary numbers. Referring to the table above, we now see that the base ten number "512" could be represented by a single "1" transistor followed by nine "0" transistors on a computer.

Just for fun: After understanding the previous math example, we can, in a matter of speaking, say that the integers 2 through 9 are an illusion, and therefore all other letters and characters, as well as everything we can possibly look at on a computer screen. Even more fundamentally, even 1 and 0 hide us from the truth of On and Off; electrons or the absence thereof. As odd or even uncomfortably humorous as this may sound, it is nevertheless true, and explained in another section. It is necessary to say, however, that this does not prove or even support postmodern theory, if you know and care to make that connection. We see that computers are inextricably bound to the binary system, thus making different base number systems required reading for any serious computer user or designer. This knowledge is not limited to computer-related fields, however. Geneticists deal extensively with base four in their research, as the four possible nucleotide combinations fulfill that system (yes, there are five nucleotides, but only four will be found in a strand of RNA or DNA). And the applications don't end there either.

Number Bases: Introduction & Binary Numbers | Purplemath Number Bases: Introduction & Binary Numbers Intro & Binary Base-4 & Base-7 Octal & Hex Purplemath Converting between different number bases is actually fairly simple, but the thinking behind it can seem a bit confusing at first. And while the topic of different bases may seem somewhat pointless to you, the rise of computers and computer graphics has increased the need for knowledge of how to work with different (non-decimal) base systems, particularly binary systems (ones and zeroes) and hexadecimal systems (the numbers zero through nine, followed by the letters A through F). Content Continues Below MathHelp.com In our customary base-ten system, we have digits for the numbers zero through nine. We do not have a single-digit numeral for "ten". (The Romans did , in their character "X".) Yes, we write "10", but this stands for "1 ten and 0 ones". This is two digits; we have no single solitary digit that stands for "ten". Instead, when we need to count to one more than nine, we zero out the ones column and add one to the tens column.

When we get too big in the tens column -- when we need one more than nine tens and nine ones ("99"), we zero out the tens and ones columns, and add one to the ten-times-ten, or hundreds, column. The next column is the ten-times-ten-times-ten, or thousands, column. And so forth, with each bigger column being ten times larger than the one before. We place digits in each column, telling us how many copies of that power of ten we need. Affiliate The only reason base-ten math seems "natural" and the other bases don't is that you've been doing base-ten since you were a child. And (nearly) every civilization has used base-ten math probably for the simple reason that we have ten fingers. If instead we lived in a cartoon world, where we would have only four fingers on each hand (count them next time you're watching TV or reading the comics), then the "natural" base system would likely have been base-eight, or "octal". Binary Advertisement Let's look at base-two, or binary, numbers. How would you write, for instance, 1210 ("twelve, base ten") as a binary number?

You would have to convert to base-two columns, the analogue of base-ten columns. In base ten, you have columns or "places" for 100 = 1, 101 = 10, 102 = 100, 103 = 1000, and so forth. Similarly in base two, you have columns or "places" for 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, and so forth. The first column in base-two math is the units column. But only "0" or "1" can go in the units column. When you get to "two", you find that there is no single solitary digit that stands for "two" in base-two math. Instead, you put a "1" in the twos column and a "0" in the units column, indicating "1 two and 0 ones". The base-ten "two" (210) is written in binary as 102.

A "three" in base two is actually "1 two and 1 one", so it is written as 112. "Four" is actually two-times-two, so we zero out the twos column and the units column, and put a "1" in the fours column; 410 is written in binary form as 1002. Here is a listing of the first few numbers: ← swipe to view full table → decimal Then 1011001012 converts to 35710. Converting decimal numbers to binaries is nearly as simple: just divide by 2. Convert 35710 to the corresponding binary number. To do this conversion, I need to divide repeatedly by 2, keeping track of the remainders as I go. Watch below: The above graphic is animated on the "live" web page. As you can see, after dividing repeatedly by 2, I ended up with these remainders: These remainders tell me what the binary number is. I read the numbers from around the outside of the division, starting on top with the final value and its remainder, and wrapping my way around and down the right-hand side of the sequential division.

Then: 35710 converts to 1011001012. WyzAnt Tutoring Affiliate This method of conversion will work for converting to any non-decimal base. Just don't forget to include that first digit on the top, before the list of remainders. If you're interested, an explanation of why this method works is available here . You can convert from base-ten (decimal) to any other base. When you study this topic in class, you will probably be expected to convert numbers to various other bases, so let's look at a few more examples ... URL: Numeral Systems - Binary, Octal, Decimal, Hex Numeral Systems n - can start from negative number if the number has a fraction part. N+1 - the number of digits Binary Numeral System - Base-2 Binary numbers uses only 0 and 1 digits. B denotes binary prefix.

Examples: 101012 = 10101B = 1×24+0×23+1×22+0×21+1×20 = 16+4+1= 21 101112 = 10111B = 1×24+0×23+1×22+1×21+1×20 = 16+4+2+1= 23 1000112 = 100011B = 1×25+0×24+0×23+0×22+1×21+1×20 =32+2+1= 35 Octal Numeral System - Base-8 Octal numbers uses digits from 0..7.

Examples: 278 = 2×81+7×80 = 16+7 = 23 308 = 3×81+0×80 = 24 43078 = 4×83+3×82+0×81+7×80= 2247 Decimal Numeral System - Base-10 Decimal numbers uses digits from 0..9. These are the regular numbers that we use. Example: 253810 = 2×103+5×102+3×101+8×100 Hexadecimal Numeral System - Base-16 Hex numbers uses digits from 0..9 and A..F. H denotes hex prefix.

2816 = 28H = 2×161+8×160 = 40 2F16 = 2FH = 2×161+15×160 = 47 BC1216 = BC12H = 11×163+12×162+1×161+2×160= 48146 Numeral systems conversion table ceafinney: names of bases for number systems ceafinney > miscellania > bases Names of bases for number systems A base-2 number system is called binary and a base-10 system decimal, but what how does one term number systems based on 3, 4, 5, 8 or 12 ? This is a question that I had to answer in 1999 March, so I did some research, and below are (almost all of) my findings. Please inform me of errors that I have made or of additions that you would like to make via email . Base Most commonly “octal” but also “octonal” or “octimal”.

[C] “hexadecimal” is the common computer-science terminology, but it is unsatisfactory because it is a combination of the Greek “hexa” and the Latin “decim”. The proper Latin should be “sedecim” or “sexdecim”, yielding either “sedecimal” or “sexadecimal”. Schwartzman writes: “Since hexadecimal is a rather long word, it is sometimes abbreviated hex. The word hexadecimal is unusual because Greek and Latin elements are combined; the expected purely Latin form would be sexadecimal, but then computer hackers would be tempted to shorten the word to sex.” [emphasis added] References The Words of Mathematics: an etymological dictionary of mathematical terms used in English (ISBN 0-88385-511-9). Base converter Base converter This simple tool allows you to convert a same number between its representations under different number systems. For example, the number is equal to 3.14159... in a decimal system (base 10), or to 3.243F6... in a hexadecimal system (base 16), or to 11.001001... in a binary system (base 2).

Enter your number to write this number under base Computational precision (decimal places): =3.14159..., e for e=2.71828..., sqrt(2 for 2 = 1.41421... Please use capital letters in the representations of base >10. The algebraic operations + - * / ^ are allowed, as well as pairs of parentheses. Accepted transcendental functions are sin(), cos(), exp(), log(), atan(). This page is not in its usual appearance because WIMS is unable to recognize your web browser. In order to access WIMS services, you need a browser supporting forms. In order to test the browser you are using, please type the word wims here: and press ``Enter''. Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program. Description: converts a number between different numbering systems, arbitrary precision.

This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games Keywords: wims, mathematics, mathematical, math, maths, interactive mathematics, interactive math, interactive maths, mathematic, online, calculator, graphing, exercise, exercice, puzzle, calculus, K-12, algebra, mathï¿½matique, interactive, interactive mathematics, interactive mathematical, interactive math, interactive maths, mathematical education, enseignement mathï¿½matique, mathematics teaching, teaching mathematics, algebra, geometry, calculus, function, curve, surface, graphing, virtual class, virtual classes, virtual classroom, virtual classrooms, interactive documents, interactive document, algebra, number, number, number base, convert, decimal, binary, octal, hexadecimal Hex to Decimal Converter to Convert Base 16 to Base 10 [ Skip to Calculator ] The Hex to Decimal Converter on this page will convert base 16 numbers to base 10 numbers, and show its work. This free online hexadecimal to decimal calculator will convert hexadecimal numbers into decimal numbers and display a conversion chart to show how it arrived at the answer.

If you're not sure what a hexadecimal number is, or you would like to convert a decimal number into a hexadecimal number, please visit the Decimal to Hex Conversion page. What Are Hexadecimals and Why Are They Used? Hexadecimals are numbers that use the base 16 system of counting and expressing value. Instead of using just the numeric characters 0-9 like we're accustomed to, hexadecimals also use the letters A-F to represent the values 10-15. This allows us to more succinctly express or communicate numeric values than other number systems that have fewer numeric characters to work with (decimal, octal, binary). To illustrate, suppose you were a computer programmer and you needed to write or communicate a binary number like 1101101101102 (the numbering system recognized by computers). The decimal equivalent of that binary number would be "3510" whereas the hexadecimal equivalent would simply be "DB6". So as you can see, the primary reason we use hexadecimals instead of other number systems is simply because they are more compact and therefore more convenient to write and communicate.

How to Convert Hexadecimal to Decimal If the hexadecimal number you are converting is less than 16, then you can simply use the following hex to decimal conversion chart: Conversion Chart for Hexadecimals Less Than 16 Base 16 :

Question: What name is given to a number system with base sixteen?
Answer: HEXADECIMAL