Base of a number

The 'base' of a number system is the number of symbols used to count with....

Base 10 - decimal

We are used to BASE 10 as we have ten different number symbols that we use to represent all number values: 0123456789.

This is thought to be because we have ten fingers to count with!

Base 10 is called decimal because of the Latin word for number 10 - decem

We count like this:

Fuzzles to count...	Decimal Number	How we count....
	0	Start at 0
	1	first we have 1
	2	then 2
		etc
	9	Up to 9
	10	Start back at 0 again, but add 1 on the left
	11
	12
		etc.
	19
	20	Start back at 0 again, but add 1 on the left as we now have two lots of 10
	21	And so on!

The far right digit is the number of units, the second from right the number of 10s, the third from right the number of 100s... you did this at Primary School.... HTU

The number 345 is:

(3 x 100) + (4 x 10) + (5 x 1)

Another way of expressing this is:

(3 x 10²) + (4 x 10¹) + (5 x 10⁰)

In this way we can see each column of numbers relates to the base number to a power.

Base 2 - Binary

Computers use BASE 2 becase circuit outputs are either on or off (at operating voltage or at zero voltage) - two states - binary.

They use two symbols 0 for off and 1 for on.

Fuzzles to count	Binary Number	How we count
	0	Start at 0 - nothing to count!
	1	Then 1 in our units column
	10	When we add another we have to start back at 0 again, but add 1 on the left (like when we get to 10 in decimal)
	11	Three gives us one full binary row and one extra....
	100	Four has us having to move to another column and reset the other two to zero
	101	Five means the far right hand one now has a one
	110	Adding another makes that end digit become a zero and the column to the left become a one.
	111
	1000	Start back at 0 again (for all 3 digits), add 1 on the left
	1001	And so on!

Our binary numbers columns relate to the base number raised to a power, just like the decimal ones did....

We can work out the value in decimal by doing the following:

1001 = (1 x 2³) + (0 x 2²) + (0 x 2¹) + (1 x 2⁰)

1001 = 8 + 0 + 0 + 1