Lots of good explanations here, but one thing I'd like to clarify. WHY we add digits together to represent larger numbers. Understanding this helped me to count in binary when I was a young IT technician.

In base 10, we have 10 numbers we use to count everything, each represented by a single digit 0-9. There is no single digit to represent the number 10, so we add a digit to the left and start over at 0 on the right. Hence, the number 10. Then 11-19 in serial.

But we've run out of digits to use again. So we add another digit to the left and start over on the right. Thus, 20.

When you get to 100, you're now starting over at the right-most digit and have to fill up both right digits before the left digit moves up one.

Same goes for binary, where the only two digits are 0 and 1. Once you've counted to one, you've run out of digits to use, so you add a 1 to the left and start over on the right. So 2 is written as 10 in binary. Then 3 is 11. Then you've run out of digits again, so you add another one to the far left and start over. 4 is 100. 5 is 101. 6 is 110. 7 is 111. No more space, so add another 1 to the left and start over. 8 is 1000. 9 is 1001. 10 is 1010. 11 is 1011. 12 is 1100. And so on...

With computers, we sometimes use a hexadecimal numbering system, also known as base 16 (hex = 6, deca = 10). In this case, we need 16 unique digits before we start reusing them. So we borrow from the alphabet. We use 0-9, then go through A-F before we add a 1 to the left and start over at 0.

You can literally create a base-anything and use that to count numbers. Once you figure out how we add digits to count, it's super easy!