Converting between number bases using binary - Fundamentals of data representation - AQA - GCSE Computer Science Revision - AQA - BBC Bitesize (2024)

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Converting between number bases using binary Binary to decimal Question Decimal to binary Question

Converting between number bases using binary

Binary to decimal

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

128

128
64
32
16
8
4
2
1

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

Example - binary number 1111100

128	64	32	16	8	4	2	1
0	1	1	1	1	1	0	0

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

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Question

What would these binary numbers be in decimal?

1001
10101
11001100

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.

128	64	32	16	8	4	2	1

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.

128	64	32	16	8	4	2	1
	1

128
64	1
32
16
8
4
2
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.

128	64	32	16	8	4	2	1
	1		1

128
64	1
32
16	1
8
4
2
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.

128	64	32	16	8	4	2	1
	1		1		1

128
64	1
32
16	1
8
4	1
2
1

Place a 0 in each remaining empty place value column.

128	64	32	16	8	4	2	1
0	1	0	1	0	1	0	0

128	0
64	1
32	0
16	1
8	0
4	1
2	0
1	0

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 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.

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