Number Systems and Bases

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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 310 to 51″ and 456 means “4100 + 510 + 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!

Kalid Azad loves sharing Aha! moments. BetterExplained is dedicated to learning with intuition, not memorization, and is honored to serve 250k readers monthly.

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60 Comments

  1. hey, this is a wonderful site, its really helped me in my assignment in college, but there is a question that i can’t answer and that is, Why do we use different number bases

  2. Hi Alex, good question. I think we use binary (base 2) in computer systems because they are the easiest to build. It’s easier to make a switch that turns on or off (1 or 0) rather than one that has to go between 10 states.

    For our choice of base 10, it’s probably because we have 10 fingers. Though some ancient civilizations used base 60, base 10 is pretty natural to us as we count off items on our hands.

  3. i want to know the history of the number system o to 9, when and why did it start

  4. Negative bases can be used, too. For example, base-negative-10. 0-9 are as in base-10, but 10[-10] is -10, 11[-10] is -9, …, 99[-10] = 81, 100[-10] = 100 (because -10 x -10 = 100). Seems silly, but I actually read a paper once in which using base-negative-two was useful for reducing the size of some electronic circuits.

  5. Hi kalid, could you elaborate a bit more on binary system,.me being less than average, need a bit more explanation and examples.
    thanks
    sudarshan

  6. @Sudarshan: Thanks for the comment — I might need to write a separate article on binary. The closest analogy I can think of it that binary is an odometer that “clicks over” when it reaches two — that is, you get 0, 1, then 10 (since you need to go the next digit).

    @Jeet: Glad you enjoyed it.

  7. Here’s an interesting pop quiz question for all you who just finished reading this page: which is a more powerful base for division, base 10 or base 16? Hint: The lesson on prime numbers will shed some light.

  8. About 50 years back the currency system in India was base 16 system. One INR was divided by 16 to be called as one “paisa” but now , one paisa=1/100 INR

  9. I just have to say, thank you! I have been researching all over the internet trying to understand the various Number Base Systems and your description just spelled it out for me. I think that all too often, Math experts do not know how to relate math concepts to the average person. Your clear and consise description was put into terms that I could understand. Thanks a lot.

  10. Well I really think we live in a base 9 system if you ask Me. Consider this : 0 is a non number and you don’t get 9 until 9 is complete so the turn over is at the end of nine and when 10 starts it is just a fraction until 10 is complete which is really just a one again. so the end is at the end of nine or when we actually have nine in possession. so we Have nothing with a zero so that is not a number it is nothing at all so we start with 1 and proceed 23456789 then we have 10 11 12 13 14 15 16 17 18. then we have 19 20 21 22 23 24 25 26 27. Now what emerges is the true meaning of the number equation that is all the ones are lined up with the ones all the 2s with the 2s 3s with the 3s and so on now we consider the brake down by 3 or the square root of 9. Lets take a looksy
    ‘A’B’C
    ’1’2’3
    ’4’5’6
    ’7’8’9

    ‘A”B”C
    10 11 12
    13 14 15
    16 17 18

    ‘A”B”C
    19 20 21
    22 23 24
    25 26 27

    Tesla would be proud of my thinking I think
    now consider A+B=C right 1+2=3 or 4+8=12 right you can even go a step farther and say A1+B2=C3 or 10+11( for 11 is 2 by the power of the nine and 10 is a 1)=21 or just another 3 or more accurately a multiple of 3 so to say line A + line A = multiple of 3 is accurate and more specific each position on the line added to a specific number on another line has a special relationship with the position of the end results so all in all there is nine numbers and zero is no number and 10 is like the octave in music or the same as 10 or to say 10 is the begging of the 2nd unit congruent with 1

    Pretty tricky if you think about it. Yeah AxC is always C this is true. Check it for your self and see I am not lying. AxA=A BxB=A

    Now these laws become real important when we try to understand the mechanics of Riemann’s non trivial zeros and the true meaning of prime numbers and how they interact in the whole scheme of things, Like for example how all prime numbers squared can only be in the 1,4,7, line. It is dictated by the law AxA=A and BxB=A yet CxC can only be a C for it is always in the 3,6,9,line by the fact of it being a multiple of 3.
    So good luck and good fortune to all that read this. Long live the 3,5,7 and the 1,5,9 for it can show us even more wonders if you let it

    Mikel

  11. I think if we were all used to using base 2, we will be good in maths since is simpler but I guess no one really up to learning a new system huh

  12. Um, in ternary notation, I heard that “9″ is written as “100″ but “6″ is written as “20″. Why did that big jump in numbers occur?

  13. @Judy, “20″ and “100″ are symbols. You are thinking in terms of their decimal values, that’s why the skip seems large. In Binary, we have 0, 1, 10, 11, 100. Now see the skip between 11 and 100, its even bigger than ternary. To ease the confusion, their complete representation is (20)base3 and (100)base3. This is different from (20)base10 and (100)base10.

  14. You might also mention that hexadecimal originated as a convenient way to condense binary into groups of four digits. When I teach people what binary and hex are, I start with binary, and at some point start grouping the digits into fours, and then explain that these 16 combinations could be represented with a single character, and hence hex was born. (If introducing octal, I group in 3′s first, because people are already familiar with octal’s digits.)

  15. @Anon: Great point. Yep, hex is a good shorthand, and oftentimes I have to explicitly separate out the hex digits to see what’s happening with a bitmask.

  16. Hi, do you know what is a unitary system?..i have a report on that and all the search results just point to a government system. Is unitary system the same with unary?.. I will really appreciate a reply.thanks

  17. This is great. Very helpful. Hope you’ll post more of these wonderful lessons so that can stop biting my nails due to the frustration.hahaha…

    -Joy

  18. @randy: Base 4 would have the numbers 0-3. So it’d count 0, 1, 2, 3, 10 (4), 11 (5), 12 (6), 13 (7), 20 (8) and so on.

  19. I appreciate the information on this website, but could you possibly help me further? I have a midterm and my teacher expects us to be able to write a table for either addition or multiplication for a base (seven, five, sixteen, etc…) in order to answer problems that will be on the midterm. Thank you in advance for your help!

  20. I drop a leave a response when I like a post on a website or I have something to contribute
    to the conversation. It’s a result of the passion communicated in the article I looked at. And on this post Number Systems and Bases | BetterExplained. I was actually moved enough to drop a comment :) I do have a few questions for you if it’s okay.
    Could it be simply me or does it appear like some of these comments come across as if
    they are left by brain dead people? :-P And, if you are posting at other social sites,
    I’d like to follow you. Could you make a list all of all your social pages like your twitter feed, Facebook page or linkedin profile?

  21. Off topic, but could you do a article on one’s and two’s compliment, or even ten’s compliment because that stuff is confusing and I would love a easy visualization to that

  22. thank you for giving a such a valuable information and i have a doubt that why object codes are stored in hexa decimal number system in c programming

  23. Thanks for this article. I am doing research on the Babylonian sexagesamil (60 base) time system. I was struggling to understand what a base system was. All the articles kept referring to it without fully explaining. I will be adding this as a Math lesson for my students to play with in addition to their regular Math lessons. I am excited to show them another way that History and Math are linked together!!
    Quick question: I didn’t look, but did you do that separate article about the binary system as asked in the comments? Could you save me a little time and post the link? Thanks so much. I greatly appreciate it.

  24. Sincerely speaking, i seem nt 2 understand d idea in “tick over”. Also, can u pls help me wit d names f d system and their representation, eg Base 2 – Binary – 0,1. Thnks in advance.

  25. Sincerely speaking, i seem nt 2 understand d idea in “tick over”. Also, can u pls help me wit d names f d system and their representation, eg Base 2 – Binary – 0,1. Thanks in advance.

  26. For those who need a bit more help,
    Think of it this way:
    If you count to ten using you fingers, the first finger to come up represents 1. If you continue counting, you will reach 10, which represents a complete set; you have no other fingers to continue counting.

    In order to continue, you must clench both hands into fists in order to start again with the next set, ie. 11, 12, 13… . This is why it is called the Base 10 system and not the Base 9 system.

    The number 10 represents a complete set of the base 10 system. When you count 7,8,9- the next number is the final, finishing touch to the set. It is 1 and 0, meaning 1 set, and 0 more, 10. If we continue, it continues saying, “1 set and 1 more,” “1 set and 2 more,” eventually reaching 19, which should continue into “2 sets, and 0 more.”

    Also, 0 is a number, it is in fact even. Picture this:
    Imagine a scale, when it has nothing on it, the scale is perfectly balanced, even.
    Now, place 1 item on either side, it is now uneven (odd).
    Now, place 1 more item equal in weight (2 total) on the opposing side, it is even again.
    If you reached 9 objects, the scales would be uneven, the 2 sides would have different amounts. When you look at both your hands, you should have an even number of fingers in total, 5 and 5 making an even 10.

    —-For those who want more, I’m moving onto Base 12—-
    This is my favourite of all the bases. There are many reasons why I love Base 12, but one of the easiest to see is simple division.
    You see, when we like to divide, we like simple numbers, not decimals, only whole (2, not 1.6666…).
    Carpenters know this very well, since they use the imperial system (it has problems of it’s own, but it does use base 12).
    Imagine this:
    Base 12: put down 12 objects on the table, put it into 2 equal piles.
    how many do you have in each? 6.
    Now put down 12 objects again, put it into 4 equal piles.
    How many? 3.
    Put down the 12 one last time, cut them into 3 equal piles.
    How many now? 4.

    Base 10: put down 10 objects on the table, put it into 2 equal piles.
    how many do you have in each? 5.
    Now put down 10 objects again, put it into 4 equal piles.
    How many? You can’t you need to cut 2 of them in half, making 4 halves, then distribute them amongst all 4, making each pile 2 and a half, or 2.5, ewwww.
    Put down the 10 one last time, cut them into 3 equal piles.
    How many now? This one is insanity, you have to cut 1 of them into 3 equal pieces, which is really difficult to do since you have to use angles to figure out the equally distributed, 120 degree slices, how much is a third again? oh yeah its .33333333… repeated, you can never have a perfect number to work with, it’s always going to be wrong, with 10 cut into 3 piles, you end up with 3 and 1 third, or 3.3333333333… .

    Anyways, that’s just division, if you are interested in a diagram of why I love base 12, just look at this picture I made: http://prntscr.com/3wladb
    I really hope humanity educates ourselves to base 12, it would be beautiful.

  27. Just reading your piece on number bases and understand most of it except this which makes no sense.

    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 a

    In school I had no problem and had great fun messing with number bases but the above????

    Regards

    Frank

    PS I’ve returned to maths!

  28. Hi Frank! Glad to hear you’re getting back into it :).

    I realize the post was mis-formatted and all the numbers appeared on one line. I’m surprised you were able to make sense of even the earlier ones!

    It should be fixed now, and should read:

    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)

  29. Yep, that’s great. Thanks Khalid.

    One thing I’ve noticed with Maths that the study of it is not linear. I’m studying several maths subjects at the same time and it’s interesting the way it’s all linked together.

    For instance set theory seem to be in practically all branches of maths and there are other crossovers with other subjects as well. This of course is pretty obvious when you think about it but only when you think about it.

    The way maths was taught in school was awful. Maybe maths should be taught as one subject so to speak – meaning that you learn what you need to know as you progress so one moment you’re doing algebra, the next set theory, then trig, then geometry, then calculus and so on.

    To me that would seem the logical way to do it.

  30. Base-12 (dozenal) is a personal fave of mine too! It’s down to that or base-18. Base-18 has the same overall kind of advantages to it as dozenal, plus the awesome 3-digit rotational multiples of 7 (just like decimal multiples of 27 and 37). That’s because 7 cubed equals 111 (base-18). Decimal 111 is 3 times 37; 999 is 27 times 37.

    Decimal has 037, 370, 703, 259, 592, 925, etc. Base-18 has 02D (D the digit for 13), 2D0, D02, 083, 308, 830, etc.

    Dozenal 101 is composite, meaning its dozenal factors 5 and 25 have rotational multiples in 4 digits – because they’re factors of 1,111. But I don’t find dozenal with that facet in 3 digits, because its 111 is prime (decimal 157). Base-18 has it both in 3-digit and 4-digit because its 101 (and thus 1,111) is divisible by 5 and its square (17) and by D.

    Dozenal is rhythmic in its dozens with its multiples of 3, 4, and 6, but not 9. That is to say, multiples of 3 have for the units’ place a 3, 6, 9, or 0; for multiples of 4, it’s 4, 8, or 0. Base-18 is rhythmic too, not in 2-squared (4) but in 3-squared (9). Multiples of 3 have as unit digit 3, 6, 9, C, F, 0; for multiples of 9, either 9 or 0.

    I guess just whichever is easier to use and/or whichever has more cool stuff.

  31. Hey thanks Kalid. I think that this is an awesome explanation of something I was really struggling to understand :)

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