Converting decimal (denary) into binary is very simple and can be done by simply remembering the following numbers:
128, 64, 32, 16, 8, 4, 2, 1
Have you noticed anything about these numbers?
From right to left they are doubling every time!
So why do we need to remember these numbers?
Let me show you with an example.
Example 1.
Convert denary number 198 into binary.
All we need to do to convert this denary number is draw out a table. Put the numbers, 128, 64, 32 etc from above into the top row.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
We start from the far left and ask “can 128 be taken away from 198 without leaving a negative number?” As 198 is bigger than 128 we put a 1 under 128 and subtract 128 from 198;
198 – 128 = 70.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 1 | |||||||
| 198 – 128 = 70 Remainder 70 |
We now carry the remainder across to the next column and ask the same – “can 64 be taken away from 70 without leaving a negative number?” As 70 is bigger than 64 we put a 1 under 64 and subtract 64 from 70;
70 – 64 = 6.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 1 | 1 | ||||||
| Remainder 70 | 70 – 64 = 6 Remainder 6 |
We now carry the remainder across to the next column and ask the same – “can 32 be taken away from 6 without leaving a negative number?” As 6 is not bigger than 32 we put a 0 under 32 and carry 6 across to the next column.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 1 | 1 | 0 | |||||
| Remainder 70 | Remainder 6 | Remainder 6 |
We now carry the remainder across to the next column and ask the same – “can 16 be taken away from 6 without leaving a negative number?” As 6 is not bigger than 16 we put a 0 under 16 and carry 6 across to the next column.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 1 | 1 | 0 | 0 | ||||
| Remainder 70 | Remainder 6 | Remainder 6 | Remainder 6 |
We now carry the remainder across to the next column and ask the same – “can 8 be taken away from 6 without leaving a negative number?” As 6 is not bigger than 8 we put a 0 under 8 and carry 6 across to the next column.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 1 | 1 | 0 | 0 | 0 | |||
| Remainder 70 | Remainder 6 | Remainder 6 | Remainder 6 | Remainder 6 |
We now carry the remainder across to the next column and ask the same – “can 4 be taken away from 6 without leaving a negative number?” As 6 is bigger than 4 we put a 1 under 4 and subtract 4 from 6;
6 – 4 = 2.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 1 | 1 | 0 | 0 | 0 | 1 | ||
| Remainder 70 | Remainder 6 | Remainder 6 | Remainder 6 | Remainder 6 | 6 – 4 = 2 Remainder 2 |
We now carry the remainder across to the next column and ask the same – “can 2 be taken away from 2 without leaving a negative number?” As 2 can be taken away from 2 without leaving a negative we put a 1 under 2 and subtract 2 from 2;
2 – 2 = 0.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 1 | 1 | 0 | 0 | 0 | 1 | 1 | |
| Remainder 70 | Remainder 6 | Remainder 6 | Remainder 6 | Remainder 6 | Remainder 2 | 2 – 2 = 0 Remainder 0 |
As we now have 0 left our calculation must be complete. The remaining column must contain a 0.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 |
| Remainder 70 | Remainder 6 | Remainder 6 | Remainder 6 | Remainder 6 | Remainder 2 | Remainder 0 |
198 in binary is 11000110
Remember
Work from the left to the right – it doesn’t work the other way around!
The numbers on the top row double each time from right to left don’t write them in the wrong order.
Ensure you learn the method – you will get conversions on an exam paper and they are easy marks.