Number Bases

Base 10

We use "Base 10" every day, it is our Decimal Number System and has 10 digits:

0 1 2 3 4 5 6 7 8 9

We count like this:

Do you see how the second column keeps count of how many tens? So 21 is:

But there are other bases!

Binary (Base 2) has only 2 digits: 0 and 1

We count in binary like this:

Demonstration

See how it is done in this little demonstration (press play):

Each column left is multiplied by the base (2 in this case):

Also try Decimal, and other bases like 3 or 4.
It will help you understand how all these different bases work.

In base 2: 10 means "one-two and zero-ones":

Ternary (Base 3) has 3 digits: 0, 1 and 2

We count like this:

Quaternary (Base 4) has 4 digits: 0, 1, 2 and 3

We count like this:

Quinary (Base 5) has 5 digits: 0, 1, 2, 3 and 4

We count like this:

Senary (Base 6) has 6 digits: 0, 1, 2, 3, 4 and 5

We count like this:

Septenary (Base 7) has 7 digits: 0, 1, 2, 3, 4, 5 and 6

We count like this:

Octal (Base 8) has 8 digits

If Dogs ruled the world they might use base-8 instead of decimal:

0 1 2 3 4 5 6 7

Nonary (Base 9) has 9 digits:

0 1 2 3 4 5 6 7 8

We count like this:

Decimal (Base 10) has 10 digits:

0 1 2 3 4 5 6 7 8 9

Well ... we talked about this at the start but here it is again:

Undecimal (Base 11)

Undecimal (base 11) needs one more digit than Decimal, so "A" is used, like this:

Duodecimal (Base 12)

Duodecimal (base 12) needs two more digits than Decimal, so "A" and "B" are used:

Hexadecimal (Base 16)

Dogs might understand hexadecimal well.

It uses the digits 0 to 9, then the six letters A to F, like this:

So hexadecimal is:

0 1 2 3 4 5 6 7 8 9 A B C D E F

Vigesimal (Base 20)

With vigesimal, the convention is that I isn't used because it looks like 1, so J=18 and K=19, as in this table:

Sexagesimal (Base 60)

Sexagesimal works like clockwork!

There are no special codes, just the numbers 0 to 59, like we use with hours and minutes.

More About Bases

How to Show the Base

To show what base a number has, put the base in the lower right like this:

101₂
This is in base 2 (binary)

314₈
This is in base 8 (octal)

Summary

Some selected bases: