Humans tend to use the denary number system. This is the base 10 system that you are familiar with. However, computers work in the binary number system, which is base 2. Denary numbers must be converted into their binary equivalent before a computer can use them.
The denary system has ten symbols - 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The value of each denary place value is calculated by multiplying the previous place value by ten. For example:
10,000, 1,000, 100, 10, 1
So, the value of the number 124 in denary place values is actually:
10000 = 0, 1000 = 0, 100 = 1, 10 = 2, 1 = 4
This gives (1 × 100) + (2 × 10) + (4 × 1) = 124.
Binary to denary The value of each binary place value is calculated by multiplying the previous place value by two. The first eight binary place values are:
128, 64, 32, 16, 8, 4, 2, 1
In binary, each place value can only be represented by 1 or a 0.
To convert binary to denary, simply take each place value that has a 1, and add them together.
For example, the binary number 1111100 in binary place values is:
128 = 0, 64 = 1, 32 = 1, 16 = 1, 8 = 1, 4 = 1, 2 = 0, 1 = 0
Result: (0 × 128) + (1 × 64) + (1 × 32) + (1 × 16) + (1 × 8) + (1 × 4) + (0 × 2) + (0 × 1) = 124
Denary to binary
To convert from denary to binary, start by subtracting the biggest place value you can from the denary number, then place a 1 in that place value column. Next, subtract the second biggest place value you can, and place a 1 in the column. Repeat this process until you reach zero. Finally, place a 0 in any empty place value columns.
Example: convert denary number 84
- First set up the columns of binary place values.
128, 64, 32, 16, 8, 4, 2, 1
- 64 is the biggest place value that can be subtracted from 84. Place a 1 in the 64 place value column and subtract 64 from 84, which gives 20.
64 = 1
- 16 is the biggest place value that can be subtracted from 20. Place a 1 in the 16 place value column and subtract 16 from 20, which gives 4.
64 = 1, 16 = 1
- 4 is the biggest place value that can be subtracted from 4. Place a 1 in the 4 place value column and subtract 4 from 4, which gives 0.
64 = 1, 16 = 1, 4 = 1
- Place a 0 in each remaining empty place value column.
128 = 0, 64 = 1, 32 = 0, 16 = 1, 8 = 0, 4 = 1, 2 = 0, 1 = 0
- Result: 84 in denary is 01010100 in binary.
To check that this is right, convert the binary back to denary:
(0 × 128) + (1 × 64) + (0 × 32) + (1 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (0 × 1) = 84
Another way to convert a denary number to binary is to divide the starting number by two. If it divides evenly, the binary digit is 0. If it does not and there is a remainder, the binary digit is 1. Finally, reverse the digits and you have the correct number.
Remember, when coverting binary numbers, try to create a table based on the binary values.