Chapter 5 · Watch, then practise
Computer Arithmetic
Use fixed-width binary representations and follow the hardware work behind arithmetic.
3 questions · 3 with related videos. Matches are based on playlist titles; broader background matches are labeled.
What to study
- Two’s complement
- Signed overflow
- Binary multiplication
- Carry lookahead
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Notes
Representations of Binary Numbers
Neso Academy · 17:13
Choose a video · 2 lectures
Supplementary lessons on binary representation and overflow; the eight-bit calculation remains in the answer.
1. Representation and overflow
Represent −7 in eight-bit two’s complement. Does 100 + 50 fit the same signed format?
Seven is 00000111. Invert and add one to obtain 11111001, representing −7. The eight-bit signed range is −128 through 127, so 150 does not fit. Binary addition produces 10010110, interpreted as −106 in this format: signed overflow occurred. A carry-out bit alone does not diagnose signed overflow.
Binary Multiplication
Neso Academy · 11:13
Covers binary multiplication rather than treating the question as a Booth-algorithm exercise.
2. Partial products
Compute unsigned 1011₂ × 0101₂ using shifted partial products.
The multiplier has set bits in positions 0 and 2. Add 00001011 and 00101100 to get 00110111₂ = 55. This matches 11 × 5. Retain enough result bits: two unsigned n-bit inputs can require 2n bits. A signed multiplier must additionally handle sign contributions correctly.
Carry Look Ahead Adder (CLA): Parallel Adder Basics and Design | COA
Engineering Funda · 19:09
Supplementary carry-lookahead lesson for parallel carry generation.
3. Faster carries
How does carry lookahead avoid waiting for every preceding full adder?
For bit i, let gᵢ = aᵢbᵢ and pᵢ = aᵢ XOR bᵢ. Then cᵢ₊₁ = gᵢ OR (pᵢ AND cᵢ). Expanding or grouping these relationships allows carries to be calculated through a logic tree. This reduces carry-chain delay at the cost of extra logic and wiring.