1.1c Binary Arithmetic & Shifts — Cambridge IGCSE CS 0478

Binary Addition

Binary addition uses the same column-by-column method as denary addition but with only two digits. The rules are:

A	B	Sum	Carry
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

Example: Add 00110101 (53) and 00101001 (41):

Result: 01011110 = 90 ✓

Overflow

Overflow occurs when the result of an arithmetic operation is too large to be stored in the available number of bits. For 8-bit unsigned binary, the maximum value is 255. If a result exceeds this, a carry beyond the 8th bit occurs — this bit is lost, causing an incorrect result.

Example: 11000000 (192) + 10000000 (128) = 100000000 = 256, but stored in 8 bits as 00000000 — overflow error.

Exam tip: State that overflow occurs when the result of an addition or shift operation cannot be stored in the available number of bits, and the carry bit is lost. Cambridge may ask you to identify when overflow has occurred from a worked example.

Logical Binary Shifts

A logical shift moves all bits left or right by a specified number of positions. Vacated positions are filled with 0s. Bits shifted out are lost.

Logical left shift by 1: equivalent to multiplying by 2
Logical right shift by 1: equivalent to integer division by 2

Original	Operation	Result	Denary
00001010 (10)	Left shift by 2	00101000	40 (×4)
00010100 (20)	Right shift by 2	00000101	5 (÷4)

Arithmetic Binary Shifts

An arithmetic shift preserves the sign bit (MSB). Used for signed (two's complement) numbers.

Arithmetic right shift: the MSB is copied into vacated positions (preserves the sign)
Arithmetic left shift: same as logical left shift; zeros fill from the right

Example: Arithmetic right shift of 11001000 (–56) by 1: → 11100100 (–28). The sign bit (1) is preserved; value halved.

⚠️ Common Mistakes

Confusing logical and arithmetic shifts (arithmetic shifts preserve the sign bit)
Thinking a left shift always multiplies — if a 1-bit is shifted out, overflow occurs and the result is incorrect
Forgetting that vacated bits are filled with 0 in logical shifts

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Key points

Binary addition rules and worked examples
Overflow — when and why it occurs
Logical left and right shifts: multiplying and dividing by powers of 2
Arithmetic shifts and sign bit preservation

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Worksheet — Binary Arithmetic & Shifts

5 questions · 10 marks

Q1Add the binary numbers 01001101 and 00110110. Show your working and state the result in denary.[3]

✅ Mark scheme

01001101 (77) + 00110110 (54) = 10000011 [2]; denary = 131 [1]

Q2Explain what overflow means in the context of binary arithmetic.[2]

✅ Mark scheme

Overflow occurs when the result of an arithmetic operation is too large to be stored in the available number of bits [1]; a carry is generated beyond the MSB which is lost, giving an incorrect result [1]

Q3Apply a logical right shift of 2 to 01101100. State the result and its denary value.[2]

✅ Mark scheme

00011011 [1]; denary = 27 [1] (original was 108, ÷4=27)

Q4What is the effect of a logical left shift by 1 on the value of a binary number?[1]

✅ Mark scheme

It multiplies the value by 2 [1] (provided no overflow occurs)

Q5State the difference between a logical shift and an arithmetic shift.[2]

✅ Mark scheme

In a logical shift, vacated bits are always filled with 0 [1]; in an arithmetic right shift, the sign bit (MSB) is copied into vacated positions to preserve the sign of the number [1]

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Binary Addition

Overflow

Logical Binary Shifts

Arithmetic Binary Shifts

Key points

Worksheet — Binary Arithmetic & Shifts

Mini Test — Binary Arithmetic & Shifts