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:

	0	Start at 0
•	1	Then 1
••	2	Then 2
	⋮
•••••••••	9	Up to 9
••••••••••	10	Start back at 0 again, but add 1 on the left
•••••••••• •	11
•••••••••• ••	12
	⋮
•••••••••• •••••••••	19
•••••••••• ••••••••••	20	Start back at 0 again, but add 1 on the left
•••••••••• •••••••••• •	21	And so on!

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

Tens
2

Ones
1

But there are other bases!

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

We count in binary like this:

	0	Start at 0
•	1	Then 1
••	10	Start back at 0 again, but add 1 on the left
•••	11
••••	100	Start back at 0 again, and add 1 to the number on the left... ... but that number is already at 1 so it also goes back to 0 ... ... and 1 is added to the next position on the left
•••••	101
••••••	110
•••••••	111
••••••••	1000	Start back at 0 again (for all 3 digits), add 1 on the left
•••••••••	1001	And so on!

Demonstration

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

images/number-odometer.js?mode=2

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

Ten vs 10

The word ten means this many: ••••••••••

The numeral 10 means 1 lot of our base and zero extra ones

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

Twos
1

Ones
0

When not using Base 10, try reading 10 as "one-zero" instead of "ten." It stops your brain from accidentally thinking of 10 fingers!

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

We count like this:

	0	Start at 0
•	1	Then 1
••	2
•••	10	Start back at 0 again, but add 1 on the left
••••	11
•••••	12
••••••	20	Start back at 0 again, but add 1 on the left
•••••••	21
••••••••	22
•••••••••	100	Start back at 0 again, and add 1 to the number on the left... ... but that number is already at 2 so it also goes back to 0 ... ... and 1 is added to the next position on the left
••••••••••	101	And so on!

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

We count like this:

	0	Start at 0
•	1	Then 1
••	2
•••	3
••••	10	Start back at 0 again, but add 1 on the left
•••••	11
••••••	12
•••••••	13
••••••••	20	Start back at 0 again, but add 1 on the left
•••••••••	21	And so on!

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

We count like this:

	0	Start at 0
•	1	Then 1
••	2
•••	3
••••	4
•••••	10	Start back at 0 again, but add 1 on the left
••••••	11
•••••••	12
••••••••	13
•••••••••	14
••••••••••	20	Start back at 0 again, but add 1 on the left
•••••••••• •	21	And so on!

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

We count like this:

	0	Start at 0
•	1	Then 1
••	2
•••	3
••••	4
•••••	5
••••••	10	Start back at 0 again, but add 1 on the left
•••••••	11
••••••••	12
•••••••••	13
••••••••••	14
•••••••••• •	15
•••••••••• ••	20	Start back at 0 again, but add 1 on the left
•••••••••• •••	21	And so on!

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

We count like this:

	0	Start at 0
•	1	Then 1
••	2	Then 2
	⋮
••••••	6	Up to 6
•••••••	10	Start back at 0 again, but add 1 on the left
••••••••	11
•••••••••	12
	⋮
•••••••••• •••	16
•••••••••• ••••	20	Start back at 0 again, but add 1 on the left
•••••••••• •••••	21	And so on!

Octal (Base 8) has 8 digits

Two dog paws with four toes each, totaling eight toes for base eight.

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

0 1 2 3 4 5 6 7

	0	Start at 0
•	1	Then 1
••	2	Then 2
	⋮
•••••••	7	Up to 7
••••••••	10	Start back at 0 again, but add 1 on the left
•••••••••	11
••••••••••	12
	⋮
•••••••••• •••••	17
•••••••••• ••••••	20	Start back at 0 again, but add 1 on the left
•••••••••• •••••••	21	And so on!

Nonary (Base 9) has 9 digits:

0 1 2 3 4 5 6 7 8

We count like this:

	0	Start at 0
•	1	Then 1
••	2	Then 2
	⋮
••••••••	8	Up to 8
•••••••••	10	Start back at 0 again, but add 1 on the left
••••••••••	11
•••••••••• •	12
	⋮
•••••••••• •••••••	18
•••••••••• ••••••••	20	Start back at 0 again, but add 1 on the left
•••••••••• •••••••••	21	And so on!

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:

	0	Start at 0
•	1	Then 1
••	2	Then 2
	⋮
•••••••••	9	Up to 9
••••••••••	10	Start back at 0 again, but add 1 on the left
•••••••••• •	11
•••••••••• ••	12
	⋮
•••••••••• •••••••••	19
•••••••••• ••••••••••	20	Start back at 0 again, but add 1 on the left
•••••••••• •••••••••• •	21	And so on!

Undecimal (Base 11)

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

Decimal:	0	1	2	3	4	5	6	7	8	9	10	11	12	...
Undecimal:	0	1	2	3	4	5	6	7	8	9	A	10	11	...

Letters! When a base needs more than ten digits, we need extra symbols. Here we use A, B, C and so on, but some people use different systems.

Duodecimal (Base 12)

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

Decimal:	0	1	2	3	4	5	6	7	8	9	10	11	12	13	...
Duodecimal:	0	1	2	3	4	5	6	7	8	9	A	B	10	11	...

Four dog paws with four toes each, totaling sixteen toes for base sixteen.

Hexadecimal (Base 16)

Dogs might understand hexadecimal well.

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

Decimal:	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	...
Hexadecimal:	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F	10	11	...

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:

Decimal:	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	...
Vigesimal:	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F	G	H	J	K	10	...

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.

../measure/images/clock-write.js

The main advantage is that 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, which makes it easy for us to divide up hours and minutes.

More About Bases

The Number Base is also called the Radix

How to Show the Base

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

101₂This shows that's in Base 2 (Binary)

314₈This shows that's in Base 8 (Octal)