Fundamentals of data representation - AQAConverting between number bases using binary

Binary to decimal

The binary system is base two and has just two symbols, 0 and 1. The first eight binary place values are:

To convert binary to decimal, simply take each place value that has a 1, and add them together.

Example - binary number 1111100

Result - (0 × 128) + (1 × 64) + (1 × 32) + (1 × 16) + (1 × 8) +

(1 × 4) + (0 × 2) + (0 × 1) = 124

Decimal to binary

To convert from decimal to binary, start by subtracting the biggest place value possible from the decimal number, then place a 1 in that place value column. Next, subtract the second biggest place value possible, and place a 1 in the column. Repeat this process until zero is reached. Finally, place a 0 in any empty place value columns.

Example - decimal number 84

• First set up the columns of binary place values.

• 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.

• 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.

• 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.

• Place a 0 in each remaining empty place value column.

Result - 84 in decimal is 01010100 in binary.

To check that this is right, convert the binary back to decimal:

(0 × 128) + (1 × 64) + (0 × 32) + (1 × 16) + (0 × 8) + (1 × 4) +

(0 × 2) + (0 × 1) = 84

Another way to convert a decimal 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.

This table illustrates the relationship between decimal and binary numbers, from 0 up to 255.

Binary is also used within truth tables. To find out more, see the computer systems study guide.