4.5.4a Binary Integers & Arithmetic — AQA 7517

Binary Integer Representation

Unsigned integers use pure binary with positional (place) value. For an 8-bit byte, columns are:

Bit 7	Bit 6	Bit 5	Bit 4	Bit 3	Bit 2	Bit 1	Bit 0
128	64	32	16	8	4	2	1

Range for an n-bit unsigned integer: 0 to 2ⁿ − 1. For 8 bits: 0 to 255.

Converting Between Bases

Binary → Decimal

Multiply each bit by its positional value and sum them.

Example: 10110011₂ = 128+32+16+2+1 = 179₁₀

Decimal → Binary

Repeated division by 2 — record remainders from bottom to top.

Division	Quotient	Remainder
75 ÷ 2	37	1
37 ÷ 2	18	1
18 ÷ 2	9	0
9 ÷ 2	4	1
4 ÷ 2	2	0
2 ÷ 2	1	0
1 ÷ 2	0	1

Reading remainders bottom-up: 75₁₀ = 1001011₂

Binary Addition

Four rules:

0 + 0 = 0, carry 0
0 + 1 = 1, carry 0
1 + 0 = 1, carry 0
1 + 1 = 0, carry 1
1 + 1 + 1 = 1, carry 1

Worked example

Column	128	64	32	16	8	2	1
A	0	1	1	0	1	1	0
B	0	0	1	1	1	0	1
Sum	1	0	1	0	0	1	1

A (106) + B (57) = 163 = 10100011₂. Verify: 128+32+2+1 = 163 ✓

Overflow

Overflow occurs when the result of an arithmetic operation exceeds the number of bits available. In an 8-bit unsigned system, 200 + 100 = 300, but 300 > 255 — the result wraps around or raises an error.

Overflow in addition: when a carry-out is generated from the most significant bit position.

Binary Multiplication & Shifts

Shifting binary left or right is equivalent to multiplying or dividing by powers of 2:

Operation	Effect	Example
Shift left 1	× 2	00001010 → 00010100 (10→20)
Shift left 2	× 4	00000011 → 00001100 (3→12)
Shift right 1	÷ 2 (integer)	00010100 → 00001010 (20→10)

Bits shifted off the end are lost; zeros fill in from the other end (logical shift).

Exam tip: Show all carry bits when performing binary addition — examiners look for working. Know that overflow occurs when a carry exits the MSB in unsigned addition. Bit shifts are fast multiplications/divisions by 2 — essential for optimised code. For 8-bit, max unsigned value = 255; overflow means wrapping at 256.

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Worksheet — 4.5.4a Binary Integers & Arithmetic

8 questions · instantly marked · AQA 7517 standard

Q1Convert the binary number 10110101 to decimal. Show your working.[2]

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Positional values: 128+32+16+4+1 = 181 [1]; correct working shown (e.g. table or calculation) [1].

Q2Convert the decimal number 209 to 8-bit binary. Show your method.[2]

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11010001₂ [1]; correct method shown — either repeated division by 2 or subtraction of positional values (128+64+16+1=209) [1].

Q3Perform the binary addition 10101010 + 01110111. Show all carry bits and the result in binary.[3]

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10101010 (170) + 01110111 (119) = 100100001₂ but as 8-bit: 00100001 (33) with overflow carry of 1 from MSB [1]; all carry bits correctly shown [1]; overflow identified — result exceeds 8-bit range [1].

Q4Explain what overflow is in the context of binary addition. Give an 8-bit example.[3]

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Overflow occurs when the arithmetic result is too large to fit in the number of available bits [1]; produces a carry-out from the most significant bit position [1]; example: 200 + 100 = 300, which exceeds 255 (max 8-bit unsigned), generating overflow [1].

Q5A logical left shift of 3 is applied to the binary value 00000101. What is the result in binary and decimal?[2]

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00101000₂ [1]; decimal value = 40 (5 × 2³ = 5 × 8 = 40) [1].

Q6What is the maximum unsigned integer that can be stored in a 16-bit register?[2]

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Maximum = 2¹⁶ − 1 [1] = 65,535 [1].

Q7A logical right shift of 2 is applied to 10111100. What is the result in binary? What is the decimal equivalent?[2]

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00101111₂ [1]; 32+8+4+2+1 = 47 (original 188 ÷ 4 = 47) [1].

Q8Perform the addition 00111010 + 00101101 in binary, and convert your result to decimal. Show all working.[3]

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58 + 45 = 103; binary: 01100111₂ [1]; carries correctly shown [1]; decimal 64+32+4+2+1 = 103 [1].

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Binary Integer Representation

Converting Between Bases

Binary → Decimal

Decimal → Binary

Binary Addition

Worked example

Overflow

Binary Multiplication & Shifts

Worksheet — 4.5.4a Binary Integers & Arithmetic

Mini Test — Binary Integers