Ten Kinds of People: How Counting Systems Work

1What Even Is a Number System?

A number system is just two things agreed on ahead of time: a set of symbols, and a rule for what happens when you run out of them. Once you run out of symbols in one column, you carry over into the next column — that's it, that's the whole idea, no matter which system you're in.

You ran out of single symbols at 9, so you carried into a new column — and got "10." Every system does exactly this, it just runs out at a different number.

2Place Value: Powers Hiding in Plain Sight

Here's the part that makes columns actually mean something: each column isn't just "the next one over" — it's worth a fixed multiple of the column to its right, and that multiple is always the base you're counting in. In decimal, each column is worth 10× the one before it. In binary, each column is worth 2× the one before it.

That's why hex 0x48 equals (4×16) + (8×1) = 64 + 8 = 72 — the same "H" from Lesson 1.5.

3Decimal: Probably the First System Ever

Ten symbols — 0 through 9 — and it's almost certainly no accident that we have exactly ten fingers. Count on your hands, run out at ten, and you've basically invented decimal by accident. It's likely the very first counting system humans ever settled on, purely because we happened to come with a built-in set of counters attached to our wrists.

Watch the carry happen — once at the ones place, then again once the tens place fills up too:

Fill up a column, and it resets to 0 while the next column left ticks up by one. Do that twice in a row and you watch a brand new column get born.

4Binary: Only Two Symbols, Because Switches Only Have Two States

Just 0 and 1. Computers don't have fingers, but every tiny switch inside one only has two positions — on or off — so binary is the one system computers actually run on at the deepest level. You built this by hand back in Lesson 1.

Binary runs out of symbols almost immediately — right after 1 — so it carries over constantly, far more often than decimal does:

Every single column fills up after just one step, so new columns get born constantly. That's exactly why a full byte needs eight columns to count anywhere useful.

Try it yourself — count upward one switch-flip at a time, all the way to the top of a byte:

5Hex: Sixteen Symbols, Built for Talking to Computers

0 through 9, then A through F for ten through fifteen — sixteen symbols in total. Hex exists purely as a human convenience: since a byte is 8 bits, and 8 splits perfectly into two groups of 4, one hex symbol can always stand in for exactly 4 bits. You did this by hand in the last lesson.

Hex runs out of symbols too — it just runs out later. Once you hit F (fifteen), the very next value has to carry into a new column, exactly the same idea as decimal running out at 9:

That "10" isn't ten — remember, it's just whatever comes right after you run out of symbols. In hex, that's sixteen.

6Wait — Is ASCII a Number System Too?

Good moment to draw a sharp line: decimal, binary, hex, and octal are all just different costumes for the same number — the value 72 doesn't change no matter which one you write it in. ASCII is something else entirely. It's not a counting system at all — it's a lookup table that assigns meaning to a number, saying "this particular value stands for the letter H." Number systems change how a value looks. ASCII changes what a value means.

7Octal: The One That Mostly Faded

Octal uses eight symbols, 0 through 7, grouping bits in threes instead of fours. It made great sense on some very old computers whose "word size" was 12 or 36 bits — both divide evenly by 3. But nearly every computer today is built around the 8-bit byte, and 8 does not divide evenly by 3 — you'd always have leftover bits. Hex fits a byte perfectly (4+4); octal never quite does. That mismatch is exactly why octal mostly lost out to hex.

It didn't disappear entirely, though — it's still hiding in one very famous spot:

If Gramps is ever typing commands like that in a terminal, that's octal, still quietly doing its job after all these years.

8Same Number, Different Costumes

Try it yourself: pick any number from 0 to 255, and watch it get dressed up in all three counting systems at once. It's the exact same value every time — just wearing a different outfit.

The four rows above are all the exact same value, just written differently. This last part is different in kind — it's not another costume, it's what that value happens to mean if you're using it as a keyboard letter.

9The Joke, Fully Explained

"There are 10 kinds of people in the world: those who understand binary, and those who don't." It works because "10" isn't one fixed value — it's whatever number you run out of symbols at, plus a zero. In decimal, you run out after 9, so "10" means ten. In binary, you run out after 1, so "10" means two.