# 3. Binary Number

A binary number looks like this:

101110.11001 |
---|

A binary number only consists of zeroes and ones. Binary fractions are to the right of the binary point.

**Example 1**- **Whole numbers**

5 decimal is 101 binary. The relative position of each 1 indicates the exponent of 2 to be used. (Exponent means 'raised to the power of').

Say there is a 1 at position 3, then this represent 2 to the power of 2

A 1 at position 1 represents 2 to the power of zero

So for the number 5 there is a 1 in the third position and a 1 at the 1st position. Reduce these positions by 1 and you end up with a base 2 sequence like this

1 times 2 exponent 2 is 4 decimal

0 times 2 exponent 1 is still zero

and 1 times 2 exponent zero is 1

Therefore 4 + 1 = 5. Hence 5 decimal can be represented as 101 binary

**Example 2 - Fractions**

0.125 decimal (or one-eighth) can be represented as a binary fraction 0.001

This works in a very similar way to whole number except the exponents are now fractions of 2

As there is a zero in the exponent half and exponent quarter positions, these add up to zero, but there is a 1 at the exponent one eighths position. So these add up to 0.125 decimal.

You can now combine both methods to represent a '**real number**' in binary form.

*A real number is a mathematical term that means a number having some decimal parts, like this 1224.99938*

### Using a table to convert binary to decimal

Remembering what the various exponent of 2 are can be awkward, so you can quickly form a small table like this. Just double the decimal number at each location above zero and halve the fraction going the other way.

Decimal | 64 | 32 | 16 | 8 | 4 | 2 | 1 | . |
1/2 | 1/4 | 1/8 | 1/16 | 1/32 | 1/64 | 1/128 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Binary | 1 | 1 | 0 | 1 | 0 | 0 | 0 | . |
0 | 1 | 1 | 0 | 0 | 0 | 0 |

So just add the decimal where a 1 is present. Like this

64+32+8+1/4+1/8 = 104.375

**Note:** the position of the binary point in this format is fixed. This kind of notation is called **'fixed point binary**' This format is not often used to represent decimal numbers because there are better ways of doing it. But it does explain the basis of representing a number in binary.

The better ways of representing whole numbers is to use '**integers**' and the better way of representing decimal or 'real numbers' is **floating point notation**. Both these schemes are described in over the next few pages

**Challenge** see
if you can find out one extra fact on this topic that we haven't
already told you

Click on this link: Binary number notation

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