Binary Arithmetic and Shifts
What this topic covers
Once you can convert binary, you need to do arithmetic with it and understand binary shifts. These appear on every GCSE paper and are usually worth several marks for method.
Binary addition
There are only four rules to learn:
| Sum | Result | Carry |
|---|---|---|
| 0 + 0 | 0 | — |
| 0 + 1 | 1 | — |
| 1 + 1 | 0 | carry 1 |
| 1 + 1 + 1 | 1 | carry 1 |
Add from right to left, carrying into the next column exactly like denary addition.
Example: add 00101100 + 00011010.
0 0 1 0 1 1 0 0 (44)
+ 0 0 0 1 1 0 1 0 (26)
-------------------
0 1 0 0 0 1 1 0 (70)Check: 44 + 26 = 70. ✓
Overflow
Each register (storage location) has a fixed number of bits — usually 8. If an addition produces a result that needs a 9th bit, that bit has nowhere to go. This is called overflow.
Example: 10000000 (128) + 10000000 (128) = 1 00000000. The leading 1 is the 9th bit — it is lost, so the stored answer is 00000000 (0), which is wrong.
- Overflow causes an error because the result is too large for the register.
- The computer may set an overflow flag to warn that the answer is invalid.
Binary shifts
A logical shift moves every bit left or right by a set number of places. Bits shifted off the end are lost, and zeros are shifted in to fill the gaps.
Left shift = multiply
Shifting left by 1 place multiplies by 2. Left by n places multiplies by 2ⁿ.
00000110 (6) shift left 1 → 00001100 (12)Right shift = divide
Shifting right by 1 place divides by 2 (integer division — any remainder is lost). Right by n divides by 2ⁿ.
00010100 (20) shift right 2 → 00000101 (5)Losing precision
Because right shifts throw away the bits that fall off the end, information can be lost:
00000101 (5) shift right 1 → 00000010 (2, not 2.5)The 0.5 is gone — this is a loss of precision.
Worked example
Show the effect of a logical left shift of 2 on 00000011.
1. Start value: 00000011 = 3.
2. Shift left 2: 00001100 = 12.
3. Effect: 3 × 2² = 3 × 4 = 12. ✓
Common mistakes
- Forgetting that a shift fills the empty end with zeros.
- Writing the answer with the wrong number of bits — keep it to 8.
- Saying a right shift "loses precision" only sometimes — say it loses precision whenever a 1 falls off the right-hand end.
- Confusing overflow (result too big for the register) with a normal carry within the columns.
Exam tips
- State the effect in words ("left shift by n multiplies by 2ⁿ") as well as showing the bits — questions often ask for both.
- For overflow questions, always say the extra bit is lost and the answer is therefore incorrect / an error.
- Line your columns up neatly; most lost marks are careless place-value slips.
Key facts to remember
- Binary addition carries when a column totals 2 or more.
- Overflow = result needs more bits than the register has → the extra bit is lost → error.
- Left shift by n = × 2ⁿ; right shift by n = ÷ 2ⁿ (right shifts can lose precision).