Binary, and why ten was never special
We count in tens because we have ten fingers. That is the entire reason.
Writing 247 means two hundreds, four tens and seven ones. Each position is worth ten times the one to its right.
There is no mathematical reason for ten. It is anatomy. Cultures that counted on finger joints used twelve; the Babylonians used sixty, which is why an hour has 60 minutes and a circle 360 degrees. The arithmetic works identically in any base.
What a base actually is
The base is what each position multiplies by. In base ten the positions are 1, 10, 100, 1000. In base two they are 1, 2, 4, 8, 16, 32.
Three different ways of writing down the same number of things. None is more correct. Binary needs more digits because it has fewer symbols to work with; hexadecimal needs fewer because it has more.
Why computers use two
Not because binary is elegant, but because two states are easy to build reliably. A wire carrying voltage or not; a transistor on or off; a region of disk magnetised one way or the other.
A component that had to distinguish ten voltage levels would need each level held to within a tenth of the range. Distinguishing two is far more tolerant of noise, heat and manufacturing variation. Binary is the engineering choice, not the mathematical one.
Converting, both directions
Binary to decimal: write the place values above the digits and add the ones with a 1 beneath them.
Decimal to binary: repeatedly divide by 2 and record the remainders, then read them bottom to top.
- 13 / 2 = 6 remainder 1
- 6 / 2 = 3 remainder 0
- 3 / 2 = 1 remainder 1
- 1 / 2 = 0 remainder 1
- Read upward: 1101
The same procedure works for any base - divide by the base, collect remainders. Nothing about it is specific to two.
Hexadecimal, and why colours look like that
Base 16 uses 0-9 then A-F for ten to fifteen. Its value is that one hex digit is exactly four binary digits, so hex is a compact way of writing binary that humans can read.
This is why a colour on a web page is written #FF8800: three pairs of hex digits giving red, green and blue, each from 0 to 255. FF is 255, so that is full red, 88 is about half green, 00 is no blue - an orange. Once you can read hex, colour codes stop being arbitrary strings.
Powers of two are worth knowing
- 2^8 = 256 - one byte, and why so much old software capped a value at 255
- 2^10 = 1,024 - a kilobyte
- 2^16 = 65,536 - the classic 16-bit limit
- 2^32 = about 4.3 billion - why 32-bit systems cannot address more than 4 GB
These numbers show up constantly in computing, and they look arbitrary until you know they are just where the places run out.
The Gangnam Style bug. YouTube's view counter used a 32-bit signed integer, which maxes out at 2,147,483,647. In 2014 that video passed it and the count broke. Google rewrote it to 64-bit, which tops out around 9.2 quintillion. A place-value limit, in public.
What this is really teaching
Place value is usually taught as a fact about our number system. Seeing it work in other bases reveals it as a general idea: any base, any positional system, same arithmetic underneath.
That shift - from 'this is how numbers work' to 'this is one representation of how numbers work' - is a small version of something mathematics does constantly, and it makes a lot of later material easier to accept.