We write numbers in groups of ten, and it feels like the only possible way, but it is a habit that comes from having ten fingers. Nothing in mathematics requires it.
What base ten actually means
In the number 347, each column is worth ten times the one to its right: 3 lots of 100, plus 4 lots of 10, plus 7 lots of 1.
If the columns were worth five times the one to the right, the place values would be 1, 5, 25, 125, and you would only need the digits 0 to 4. The base is simply how many you count before carrying.
Base two, or binary
| Place value | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|
| Digits for 13 | 0 | 1 | 1 | 0 | 1 |
| What it adds to | 0 | 8 | 4 | 0 | 1 |
Thirteen is written 1101 in base two, because 8 plus 4 plus 1 is 13. Only the digits 0 and 1 are ever used, because as soon as you reach the base, you carry.
The binary number 1101 is not one thousand one hundred and one. Read it as one, one, zero, one in base two, or say thirteen if you want its value.
Converting from base ten
Divide the number by the base, note the remainder, divide the answer by the base again, and carry on until you reach zero. Then read the remainders UPWARDS.
The first remainder belongs in the ones column, the rightmost place. Each later remainder belongs one column further left. So the last remainder you write is the leftmost digit, which is why you read from the bottom of your working to the top.
Converting back to base ten
Write the place values above the digits, multiply each digit by its place value, and add. For 1101 in base two the value is 8 + 4 + 0 + 1 = 13. For 243 in base five it is 50 + 20 + 3 = 73.
A wire can be carrying current or not, and a switch can be on or off. Two states are exactly what base two needs. Building a machine that reliably distinguishes ten voltage levels would be far harder than one that only tells on from off.
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