There’s something unsatisfyingly arbitrary about the number 10. When we write numbers, e.g. 1729, our notation means that we have 1 thousand (\(10^3\)), 7 hundreds (\(10^2\)), 2 tens (\(10^1\)), and 9 ones (\(10^0\)), so the number 10 is ingrained in our very writing system. And it’s not just the Arabic numeral system: Roman numerals also organize themselves around powers of 10 (such as I=1, X=10, C=100, etc.). But why 10? (Other than the fact that we usually have 10 fingers and 10 toes [why not 20 instead?].)

Our “decimal” system centers around powers of 10, but there are other equally useful systems based on other numbers. For example, we could use sums of powers of 2, and instead of needing 10 digits (0,1,…,9) we can use just 2, namely 0 and 1. For example, the number 1101 in base 2 means \(1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\), also known as 13 in base 10. This may seem inefficient, since we need more digits (or, more correctly, “bits” in the binary setting) to express the same value, but it has many benefits. Each bit has only 2 choices, and can therefore be represented by a pair of opposites such as “on/off” (e.g. in circuits) or the two magnetic polarities. The latter is how hard drive disks store information (in binary!), which is one reason this system is beloved by computer scientists. It also allows for jokes like this: “There are 10 types of people in the world: those that understand binary, and those that don’t.”

Another common base is Hexadecimal, using powers of 16 with digits 0,…,9,A,B,C,D,E,F. (A hexadecimal digit is sometimes called a “nibble”.) It is common to prefix a hexadecimal number with “0x”, so 0x3B7 means \(3 \cdot 16^2 + 11 \cdot 16 + 7 = 951\) in decimal. It also makes the phrase “I see 0xDEAD people” perfectly reasonable (how many is that?).

But there are more possibilities still! For example, the octal system uses powers of 8 and (octal) digits 0,…,7. With this we can prove Christmas and Halloween are really the same holiday: indeed, Dec 25 = Oct 31. And don’t miss the octal system’s cameo in Tom Lehrer’s New Math song.

The possibilities are endless! We can use base \(n\) for any positive integer \(n>1\) in similar ways, but we can also use negative \(n\) as well! (My favorite is base -4.) If you want to go ever further, think about what base \((1+\sqrt{5})/2\) (a.k.a. phinary) would look like! Or the related bases Fibonacci and NegaFibonacci!

*Related*

You forgot base 2i, it’s a pretty good base.