Base systems like binary and hexadecimal seem a bit strange at first. The key is understanding how different systems “tick over” like an odometer when they are full. Base 10, our decimal system, “ticks over” when it gets 10 items, creating a new digit. We wait 60 seconds before “ticking over” to a new minute. Hex and binary are similar, but tick over every 16 and 2 items, respectively.

hey, this is a wonderful site, its really helped me in my assignment in college, but there is a question that i can’t answer and that is, Why do we use different number bases

Hi Alex, good question. I think we use binary (base 2) in computer systems because they are the easiest to build. It’s easier to make a switch that turns on or off (1 or 0) rather than one that has to go between 10 states.

For our choice of base 10, it’s probably because we have 10 fingers. Though some ancient civilizations used base 60, base 10 is pretty natural to us as we count off items on our hands.

[…] And prime numbers are prime in any number system. “1/3″ is only a repeating fraction in base 10 (.33333), and you could even argue that pi (3.14159…) is not irrational in base “pi”. But everyone can agree that certain numbers are prime and can’t be divided. You can even transmit primes in a unary number system that lacks a decimal point: […]

Negative bases can be used, too. For example, base-negative-10. 0-9 are as in base-10, but 10[-10] is -10, 11[-10] is -9, …, 99[-10] = 81, 100[-10] = 100 (because -10 x -10 = 100). Seems silly, but I actually read a paper once in which using base-negative-two was useful for reducing the size of some electronic circuits.

and there are some nice features like not having to store a separate sign bit. There’s even a number system with imaginary base, which is a bit mind-boggling:

@Sudarshan: Thanks for the comment – I might need to write a separate article on binary. The closest analogy I can think of it that binary is an odometer that “clicks over” when it reaches two – that is, you get 0, 1, then 10 (since you need to go the next digit).

Here’s an interesting pop quiz question for all you who just finished reading this page: which is a more powerful base for division, base 10 or base 16? Hint: The lesson on prime numbers will shed some light.