#numeric system

Short answer: binary has only 1 and 0 as numbers, so 1 + 1 = 10.

(Image from Binary Number System, MathIsFun.com)

Long answer:

Our arabic-numbered numeric system is positional, which means that each number can mean something different depending on the position they’re in. This is why the 1 in “1.000.000″ means a lot more than the 1 in “0.001″.

The way that these positions work is that when we run out of digits for the current position, we add a one to the digit in the next left position. So, when you are counting and you run out of numbers from 0 to 9, you add a 1 to the next position and start again (which is why 10 comes right after 9).

In binary, the only digits that you have are 0 and 1, because it’s not based in counting up to 10, but counting only up to 2. Since the base of the numeric system defines how many digits you have, 10 in base 10 is written 10, and 2 in base 2 is written 10, and 16 in hexadecimal (base 16) is written 10. They all follow the same logic: you run out of digits, one to the side and start again.

You can imagine each digit to the left as counting a “group” of the digits to the right. If you have 25 apples, you have 2 groups of 10 and 5 groups of 1. 

In binary, 10 means “1 group of 2 and 0 groups of 1″.

This is why 1 + 1 = 2 = 10₂.

Sources:

Binary number, Wikipedia