Year 8 · Spring 1 · Representations: from clay to siliconBinary addition and overflow
Topic 3 Binary, denary and hex
KS3 Computing · They're three ways of writing the same numbers. Each one uses place value; only the number of digits changes.
Computer SystemsCambridge Lower Secondary Computing 0860
You will learn to
- Use the four rules of binary addition: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 and 1 + 1 + 1 = 11.
- Add two 8-bit numbers column by column, from right to left, carrying when needed.
- Explain overflow: the answer needs a ninth bit, so the last carry is lost.
- Explain that a shift left multiplies by 2 and a shift right divides by 2.
Success criteria
- I can add two bytes and check the answer in denary.
- I can explain why 255 + 1 gives 0 in a byte.
- I can spot a sum that will overflow, because its total is more than 255.
- most significant bit
- Most significant bit: the leftmost bit, worth the most.
- least significant bit
- Least significant bit: the rightmost bit, worth 1.
- overflow
- A result too big for the bits available.
- Denary
- Base 10. Our everyday numbers.
- Binary
- Base 2. Digits 0 and 1. Columns 128 64 32 16 8 4 2 1.
- Hexadecimal
- Base 16: digits 0–9 then A–F. One digit = 4 bits.
Binary place values
★RememberStart at 1 on the right. Double each time you move left. 1, 2, 4, 8, 16, 32, 64, 128. Learn this row by heart.
The leftmost bit is the most significant bit. The rightmost is the least significant bit.
Overflow
!Watch outA byte holds at most 255. 200 + 100 = 300 needs a 9th column, which doesn't exist. The carry is lost and the byte holds 44. That is an overflow error.
e.g.ExampleAn old game score suddenly jumping back to a tiny number.
Binary shifts
- Shift left 1 place = × 2. 00010110 (22) → 00101100 (44).
- Shift right 1 place = ÷ 2. 00010110 (22) → 00001011 (11).
- 2 places = × or ÷ 4. 3 places = × or ÷ 8.
!Watch outBits pushed off the end are lost. 11 shifted right gives 5, not 5.5.
The steps for this lesson
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