The number system we generally use is a base 10 system AKA the decimal system (dec meaning ten). Ones place, tens, place, hundreds place, etc. However the choice of 10 is fairly fucking arbitrary. We could use a base 8 system (octal system), base 12 (duodecimal system), or base 2 (binary system). Here’s how they work.

## How number systems work

Any standard number system is going to have a “base” aka a “radix.” The base of a number system is how many digits are used to represent numbers. So in a base 10 number system, for example, there are 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

These digits are then organized into “places” that represent groups of numbers. I’m going to show it abstractly first and then we’ll get into some specific examples.

### The abstract: base $n$

First we have the ones place. You always gotta have a ones place. The number in this place represents single units. We could also refer to this $n^0$ since anything raised to the 0 power is 1.

Since your ones place can only hold one digit, once you use up all your digits (say you have $n$ digits in a base n number system), you need to create a second place. This place keeps track of groups of $n$. We’ll call this the $n$ place.

Once the $n$ place fills up with $n$ groups of $n$ (aka $n^2$), we need a place to store all that shit. So we’ll add another place and call it the $n^2$ place.

And so on and so forth. The next place will hold $n$ groups of $n^2$, which is $n^3$. Then we get $n^4$, $n^5$, etc. Let’s have a look at some different bases to see this shit in action.

## The base 10 number system (decimal)

The most common number system is base 10. This is your standard system with ones, tens, hundreds, thousands, etc. The reason this system has been so popular for so long is because we have 10 fingers to count on, so it just makes natural sense. There’s no inherent advantage of base 10 over other bases–if humans had 8 fingers, we’d probably be using a base 8 system instead of base 10 and it would be fine.

Pretty simple. We have 2 thousands, 1 hundred, 3 tens, and 8 ones: 2,138.

I have nothing more to say about base 10. Basic as fuck.

## Base 8 (octal)

Now things get a bit more interesting. Getting right into it. Base 8 let’s go.

The ones place stays the same… always gotta have a place to keep track of the singles. After that, since we’re in base 8, we have the $8^1$ place, which holds groups of–you guessed it–8. Next up we have $8^2$, which is 64. The 64s place. Rolls right off the tongue.

So for this number we have 4 groups of 64, 2 groups of 8, and 6 units, which ends up translating to 278 in base 10. In boojie notation we can express this as

$$426_8 = 278_{10}$$

So that’s how you do that.

## Base 12 (duodecimal)

Time for the superior number system. Base 12 is the shit because it’s divisible by 1, 2, 3, 4, and 6 which makes operations like multiplication super nice, but we’ll talk about that another time. The one wrinkle here is we need 12 different digits, while we’re only used to having 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We can just add whatever symbols we want to represent 10 and 11. Call those A and B. So our new digits are

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B

Let’s have a look at how the places behave in base 12.

So we have 5 groups of 144, 3 groups of 12, and 10 units, making 766 in base 10. Therefore:

$$53A_{12}=766_{10}$$

## Conclusion

So there you have the basics of number systems. It’s entirely based on how many digits you are using. Base 10 is cool, base 8 is interesting, and base 12 is the shit and we should all convert immediately.

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