Key Terms
- Overflow: Result too large for the number of available bits; extra bits are lost
- Carry bit: The extra bit generated when two 1s are added (1+1=10)
- Left shift: Moving all bits left by n places (multiplies by 2n)
- Right shift: Moving all bits right by n places (divides by 2n)
- Two's complement: Standard method for representing negative binary numbers
- Sign bit: The most significant bit (MSB) — indicates positive (0) or negative (1)
Must-Know Facts
- Binary addition rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 carry 1), 1+1+1=11 (1 carry 1)
- Left shift 1 place = multiply by 2; left shift 2 = multiply by 4
- Right shift 1 place = divide by 2; right shift 2 = divide by 4
- Two's complement: flip all bits then add 1
- 8-bit unsigned range: 0 to 255
- 8-bit two's complement range: -128 to +127
- Overflow occurs when carry goes beyond the MSB — extra bit is discarded
Key Concepts
- Binary addition: Work right to left, carry 1 when both bits are 1
- To negate a number (two's complement): Flip all bits, then add 1
- Left shift by n: Add n zeros on the right, drop leftmost n bits
- Right shift by n: Add n zeros on the left, drop rightmost n bits
- Overflow check: If carry out of MSB position, overflow has occurred
Common Mistakes
- Forgetting to carry when 1+1 occurs: 1+1 = 10 in binary — you write 0 and carry 1 to the next column; forgetting the carry is the most common arithmetic error
- Stopping at flip-all-bits for two's complement: Two's complement requires two steps — flip ALL bits AND then add 1 — omitting the +1 step gives the wrong answer
- Confusing left shift with multiplication: Left shift by 1 multiplies by 2, but only if no bits are lost — if the MSB is shifted out, overflow has occurred and the result is incorrect
- Not recognising overflow: If a carry bit goes beyond the available bits (e.g. a 9th bit in an 8-bit system), overflow has occurred — the result stored is wrong and this must be stated
This topic summary covers Knowledge Organiser: Binary Arithmetic within Binary Arithmetic for GCSE Computer Science. Revise Binary Arithmetic in 3.3 Data Representation for GCSE Computer Science with 15 exam-style questions and 18 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 11 of 11 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Binary Arithmetic
In binary addition, what is the result of 1 + 1?
Apply a logical left shift of 2 places to the 8-bit binary number 00010110. Give your answer as an 8-bit binary number and state its denary equivalent.