How do bases work in maths? Especially decimal ones.

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I can wrap my head around how a base of two would work.

But how would a base of 12 would work? What would come after 10?

**And how would a decimal base work?? that’s even more confusing.**

In: 0

The base is how many symbols there are per digit.

Base two means 2 symbols per digit. We pick 0 and 1 (binary)

So there’s 0 and 1, but what comes next? A new digit ’10’

In any base geater than 10, you can substitute the next symbol with a letter, for example it is customary to use the numbers 0 thru 9 as well as the letters A thru F for base 16 (hexidecimal). If you want to represent the value 16 you’ll need a second digit, ’10’.

>But how would a base of 12 work? What would come after ten?

The value would still be equal to eleven, it just might have a different name or a new single-digit symbol to represent that value. The difference would be that you could then have an eleven-teen and a twenty-eleven, the same way that you could have a, say, fourteen or twenty-four.

Or if it helps to see it, base 10 goes:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

But base 12 is:

1 2 3 4 5 6 7 8 9 X Y 10 11 12 13 14 15 16 17 18 19 1X 1Y 20

There are different systems for writing numbers – expressing a particular concept (e.g. two) in symbols (e.g. 2). Note that two and 2 are not the same thing (in how I am using them here), two is a number, while 2 is a symbol we use to express that number.

We use one particular system for expressing numbers; a decimal, or base-ten system. Another one you might be familiar with is Roman Numerals, where I is one, II is two and so on.

The system we use works on powers of ten.

So take the collection of squiggles:

> 3274

We know that it represents the number “three thousand, two hundred and seventy-four.” Breaking that down, we have:

> 3 x 1000

> 2 x 100

> 7 x 10

> 4 x 1

all added together. You might remember when learning about addition, having “units, tens, hundreds” columns etc. Each column represents a different power of ten.

And we can also go down the other way, with decimals:

> 4.078

represents:

> 4 x 1

> 0 x 1/10

> 7 x 1/100

> 8 x 1/1000

To do this, we need ten symbols to represent individual digits, as each column has ten possible options; 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. We don’t need any more symbols because the next number, ten, would be 1 ten and 0 units, so 10.

So with these ten “digits” and the simple rule of each “column” represents a different power of ten, with a point between the 10^0 = 1 column and the 10^(-1) = 1/10 column, we can write any possible number (within reason).

But why use powers of ten? We could use powers of eight instead. So our first column would still be the units column, but the next would be the eights column, and then the sixty-fourths column, and the five-hundred-and-twelves column (each column being the next power of eight).

In this system

> 3274

would mean ~~”three four-thousand-and-ninety-sizes, two five-hundred-and-twelves, seven sixty-fours and four units.” Which, would end up being the number thirteen thousand, seven hundred and sixty-four.~~ “three five-hundred-and-twelves, two sixty-fours, seven eights, and four units.” Which would end up being the number one thousand, seven hundred and twenty-four.

In this base we only need eight digits, though. 0, 1, 2, 3, 4, 5, 6 and 7. Because the next number up would be eight, which is “one eight and no units”, so 10.

As a general rule in any base system you need as many digits as the base.

Binary is a more common base. Binary is base two, so works in powers of two. So:

> 10110

would be no units, one two, one four (two-squared), no eights (two-cubed) and one sixteen (two-to-the-four). In binary we only need two digits, 0 and 1, because the next number up would be two, so “one two and no units”, i.e. 10.

Note that we can still have non-whole numbers in these other bases. We can go beyond the “decimal” point, except it would now be a “binary point” (I guess). Each column after the dot would represent a smaller power of the base. In binary 1.11 would be “one, plus one half, plus one quarter.” In base eight, 1.11 would be “one, plus one eighth, plus one sixty-fourth.”

So what if we want to go into a base higher than ten? We are going to need more digits! In base twelve we would need twelve digits, but we only have ten. We could make up completely new symbols, but traditionally we start borrowing letters. Counting in base twelve would go something like:

> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 10, 11, 12, 13…

and so on. Where A represents the number ten and B the number eleven. After 99 (which is nine one-hundred-and-forty-fours plus nine units) we would go to 9A, 9B, then A0, A1 … A9, AA, AB, B0, B1… B9, BA, BB, 100.

Similarly we could do the same after the point. So we could have a number like:

> 56.A4B

which would be “five twelves, six units, ten twelfths, four one-hundred-and-forty-fourths, eleven one-thousand-seven-hundred-and-twenty-eights.”

But we tend not to use base twelve.

Mostly you see this with base sixteen, or hexadecimal (sometimes “hex”). You might have seen colours expressed in hex, things like #8DCAA5. Essentially this is three numbers, 8D, CA and A5, but in hexadecimal, so the first number would be “eight sixteens plus thirteen units” the second “twelve sixteens plus ten units” and the third “ten twelves plus 5 units.” In decimal these would be 141, 202 and 125. The first number represents the “red”ness, the second the “green”ness and the third the “blue”ness.

The advantage of using hex is that the highest number we can get to with two digits would be FF, fifteen sixteens plus fifteen, or two hundred and fifty five. Which is more that twice as high as we can get with just two digits in decimal; 99 (ninety nine). Meaning we can cram more information into two digits; rather than having only one hundred each divisions of “red”, “blue” and “green” we can have two-hundred-and-fifty-six, giving us way, way more possible colours; in decimal we have 16,777,216 possible colours, compared with the 1,000,000 we would get if we tried to express colours in two-digit decimal.