Binary Addition and Logical Shifts

Binary addition works column by column from right to left — exactly like denary addition, but the carry threshold is 2, not 10. There are only four possible column outcomes.

Column inputs	Binary result	Digit written	Carry to next column
0 + 0	0	0	0
0 + 1	1	1	0
1 + 1	10	0	1
1 + 1 + 1	11	1	1

The fourth rule (1 + 1 + 1) matters when two operand bits are both 1 and a carry also arrives from the previous column. AQA 8525 exam questions involve at most three 8-bit numbers and will never have more than three 1s in any single column.

Write the carry row above the addition before working any column — doing it mentally causes almost all binary addition errors.

Work from the rightmost column to the leftmost, using the four rules and propagating any carry one column left.

Worked example — add 00110101 (53) and 00011010 (26):

  carry: 0 1 1 0 0 0 0 0
         ─────────────────
         0 0 1 1 0 1 0 1   (53)
       + 0 0 0 1 1 0 1 0   (26)
         ─────────────────
         0 1 0 0 1 1 1 1   (79)

Column trace (right to left):

Col	Bits	Carry in	Sum	Write	Carry out
0	1+0	0	1	1	0
1	0+1	0	1	1	0
2	1+0	0	1	1	0
3	0+1	0	1	1	0
4	1+1	0	2	0	1
5	1+0	1	2	0	1
6	0+0	1	1	1	0
7	0+0	0	0	0	0

Result: 01001111 = 64 + 8 + 4 + 2 + 1 = 79 ✓ (53 + 26 = 79)

In AQA 8525 questions, you will only need to add a maximum of three 1s in any single column, including any carry. This means the four rules from the first slide are enough.

Worked example — add 00010111 (23), 00001010 (10), and 00000101 (5):

  00010111   (23)
  00001010   (10)
+ 00000101   (5)
──────────
  00100110   (38)

Key column traces where carries propagate:

Col	Bits	Carry in	Sum	Write	Carry out
0	1 + 0 + 1	0	2	0	1
1	1 + 1 + 0	1	3	1	1
2	1 + 0 + 1	1	3	1	1
3	0 + 1 + 0	1	2	0	1
4	1 + 0 + 0	1	2	0	1
5	0 + 0 + 0	1	1	1	0

Result: 00100110 = 32 + 4 + 2 = 38 ✓ (23 + 10 + 5 = 38)