#place values

OMG zero is amazing and important guys you don’t even know.

It’s one of those things a lot of people don’t even realize had to be invented–they think, oh, it’s obvious, but it’s not as obvious as all that. Early mathematical and counting systems didn’t have zero–after all, I don’t need to count how many bags of grain I have if I have none. The idea that we need to have a symbol for nothing in order to aid in counting somethings is actually pretty counterintuitive.

You can do very basic arithmetic without zero (although it’s harder–try multiplying with Roman Numerals), but to do anything beyond that you need place values. 

Place values are essentially the idea that a digit has a different meaning depending on where it is in the number. So the 2 in 208 means something different than the 2 in 28 (200 and 20, respectively). This may seem obvious, but it isn’t true in a system like Roman Numerals. V always means 5, no matter where it is in the number. 

In base ten our place values are multiples of ten (hence the name). From right to left, 10^0, 10^1, 10^2, 10^3, etc (or going left to right from the decimal point, 10^(-1), 10^(-2), 10^(-3), etc.). We then use our ten different digits to indicate how many 1s we have, how many 10s, how many 1,000s etc. This makes it possible to have a finite number of symbols and still be able to indicate any number possible. We can the line everything up all nice and neat and add or subtract.

But to do that we need to have a symbol that means “I have no 10s (or 1s, or 10,000s), move on” and thus we arrive at zero.

This system is elegant, compact, and allows all of modern math to exist. All of it. 

Zooming out a bit, the existence of zero enables us to make precise calculations about more abstract ideas, and also leads us to ask a lot of interesting questions, which leads to more and more useful systems.

For example, someone asks: “What happens if I subtract 1 from 0?” And so we invent negative numbers. That allows us to mathematically represent owing someone something, and then we ask how negative numbers interact with positive ones, and we get debits and credits and the math of balancing your checkbook. You’re welcome, accountants.

You also can’t have computers without zero and place values, because the entire concept of electronic computing is based on the fact that it is possible to represent base two with two states: on and off. One and zero. Thus the entire basis of computer language.