Writing Binary Numbers

Binary
 

Before writing numbers in binary, let's remind ourselves of how we usually write numbers using decimal notation. Let's take the number 4302 as an example. The digit 4 in this number doesn't stand for the number 4, rather it stands for 4000, or 4 x 1000. Similarly, 3 doesn't stand for 3 but for 300 = 3 x 100, 0 stands for 0 x 10, and 2 stands for 2 x 1.

So 4302 means:

$4 \times 1000 + 3 \times 1000 + 0 \times 10 + 2 \times 1$.

Similarly, 7396 stands for:

$7 \times 1000 + 3 \times 100 + 9 \times 10 + 6 \times 1$.

What do the numbers 1000, 100, 10 and 1, which appear in these expressions, have in common? They are all powers of 10:

$1000 = 10^3$ $100 = 10^2$ $10 = 10^1$ $1 = 10^0.$

 

We can do the same with powers of 2 rather than powers of 10.

For example, the binary number 110 stands for $1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 +0 = 6$ (written in decimal notation).

And the binary number 10001 stands for $1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 16 + 0 + 0 + 0 + 1 = 17$ (written in decimal notation).

You can convince yourself that a binary number only consists of the digits 0 or 1. When you write a number as a sum of consecutive powers of 2, no other coefficients are necessary.

This sorts out the natural numbers, but what about numbers that have a fractional part?

To write a number between 0 and 1 in decimal notation, you use powers of $\frac{1}{10}$ instead of powers of 10.

Similarly, to write a number between 0 and 1 in binary, you use powers of $\frac{1}{2}$ instead of powers of 2. For example,

$0.75 = \frac{1}{2} + \frac{1}{4} = 1 \times \frac{1}{2^1}+ 1 \times \frac{1}{2^2}$.

The decimal number 0.75 is written as 0.11 in binary.

The binary number 0.1001 stands for the decimal number

$1 \times \frac{1}{2^1} + 0 \times \frac{1}{2^2} + 0 \times \frac{1}{2^3} + 1 \times {1}{2^4} = \frac{1}{2} + \frac{1}{16} = 0.5625$