Binary Addition (AQA GCSE Computer Science)

What is binary addition?

• Binary addition is the process of adding together up to three binary integers (up to and including 8 bits)
• To be successful there are 5 golden rules to apply:

Binary Addition	Binary Answer	Working
0 + 0 =	0	1s   0 = 0
0 + 1 =	1	1s   1 = 1
1 + 0 =	1	1s   1 = 1
1 + 1 =	10	2s 1s   1 0 = 2
1 + 1 + 1 =	11	2s 1s   1 1 = 3

• Like denary addition, start from the rightmost digit and move left
• Carrying over occurs when the sum of a column is greater than 1, passing the excess to the next left column

Example 1

• Add together the binary values 1001 and 0100

8	4	2	1	+
1	0	0	1
0	1	0	0
				C

• Starting from right to left, add the two binary values together applying the 5 golden rules
• If your answer has 2 digits, place the rightmost digit in the column and carry the remaining digit to the next column on the left
• In this example, start with 1+0, 1+0 = 1, so place a 1 in the column

8	4	2	1	+
1	0	0	1
0	1	0	0
				C
			1

• Repeat until all columns have a value

8	4	2	1	+
1	0	0	1
0	1	0	0
				C
1	1	0	1

• The sum of adding together binary 1001 (9) and 0100 (4) is 1101 (13)

Example 2

• Add together the binary values 00011001 and 10000100

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

• Starting from right to left, add the two binary values together applying the 5 golden rules
• If your answer has 2 digits, place the rightmost digit in the column and carry the remaining digit to the next column on the left
• In this example, start with 1+1, 1+1 = 10, so place a 0 in the column and carry the 1 to the next column

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

• Repeat until all columns have a value

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

• The sum of adding together binary 00011001 (25) and 10001001 (137) is 10100010 (162)

Example 3

• Add together the binary values 00011011, 00010110 and 00100010

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

• Starting from right to left, add the two binary values together applying the 5 golden rules
• If your answer has 2 digits, place the rightmost digit in the column and carry the remaining digit to the next column on the left
• In this example, start with 1+0+0, 1+0+0 = 1, so place a 1 in the column
• In the second column we have 1+1+1, 1+1+1 = 11, so place 1 in that column and carry the other 1 to the next column

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

• Repeat until all columns have a value

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

• The sum of adding together binary 00011011 (27), 00010110 (22) and 00100010 (34) is 01010011 (83)

What is an overflow error?

• An overflow error occurs when the result of a binary addition exceeds the available bits
• For example, if you took binary 11111111 (255) and tried to add 00000001 (1) this would cause an overflow error as the result would need a 9th bit to represent the answer (256)

256	128	64	32	16	8	4	2	1	+
	1	1	1	1	1	1	1	1
	0	0	0	0	0	0	0	1
1	1	1	1	1	1	1	1		C
	0	0	0	0	0	0	0	0