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Binary addition, overflow and shifts
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Computers add binary numbers column by column, just like you add in denary. Shifting bits left or right is a quick way to multiply or divide by 2.
Binary addition rules
0 + 0 = 0
0 + 1 = 1
1 + 1 = 0 carry 1
1 + 1 + 1 = 1 carry 1
Work from the right-hand column, carrying into the next column.
0 + 1 = 1
1 + 1 = 0 carry 1
1 + 1 + 1 = 1 carry 1
Work from the right-hand column, carrying into the next column.
Overflow
8 bits can only hold up to 255 (unsigned).
If a result needs a 9th bit, there is no room for it. This is overflow and the stored result is wrong.
11001000 (200) + 01000000 (64) = 264, which needs 9 bits.
If a result needs a 9th bit, there is no room for it. This is overflow and the stored result is wrong.
11001000 (200) + 01000000 (64) = 264, which needs 9 bits.
Binary shifts
Logical shift left: move every bit left, fill the gaps on the right with 0. Each place multiplies by 2.
Logical shift right: move every bit right, fill on the left with 0. Each place divides by 2 (losing any remainder).
Arithmetic shift right: used for signed numbers. The left gaps are filled with copies of the sign bit, so negative numbers stay negative.
00010110 (22) shifted left 2 places = 01011000 (88)
Logical shift right: move every bit right, fill on the left with 0. Each place divides by 2 (losing any remainder).
Arithmetic shift right: used for signed numbers. The left gaps are filled with copies of the sign bit, so negative numbers stay negative.
00010110 (22) shifted left 2 places = 01011000 (88)
Add 00110101 and 00011010.
- 53 + 26 in denary
- Add column by column from the right, carrying 1s.
Answer: 01001111 (79)
1 + 1 = 0 carry 1. A result too big for the bits available causes overflow. Logical shifts fill with 0s: left multiplies by 2 per place, right divides by 2. Arithmetic right shifts copy the sign bit.
The interactive lesson includes the diagrams for this topic.
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