2.1d Binary Arithmetic — Edexcel 1CP2

Binary Addition

Binary arithmetic follows the same principles as denary arithmetic, but with only two digits: 0 and 1. The addition rules are:

A	B	Sum	Carry
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1 (carry)

When adding three 1s (e.g., two 1 bits + a carry of 1): 1+1+1 = 1, carry 1.

Example: Add 00110101 + 00011010

Work from right to left, just like denary addition:

	64	32	16	8	4	2	1
A	0	1	1	0	1	0	1
B	0	0	1	1	0	1	0
Result	1	0	0	1	1	1	1

00110101 (53) + 00011010 (26) = 01001111 (79) ✓

Overflow

Overflow occurs when the result of an arithmetic operation is too large to be stored in the given number of bits.

For unsigned 8-bit numbers: maximum value is 255
For signed 8-bit two's complement: maximum positive is +127
If the result exceeds the range, the high bits are lost → incorrect result

Example of overflow (8-bit):

Adding 11000000 (192) + 11000000 (192) = 110000000 which needs 9 bits. The 9th bit is lost, giving 10000000 = −128 in two's complement — clearly wrong!

Binary Shifts

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

Shift	Effect	Example (00001010 = 10)
Left shift 1	× 2	00010100 = 20
Left shift 2	× 4	00101000 = 40
Right shift 1	÷ 2	00000101 = 5
Right shift 2	÷ 4	00000010 = 2 (remainder lost)

Left shift: shift all bits left, fill vacated positions with 0 on the right
Right shift: shift all bits right, fill vacated positions with 0 on the left
Bits shifted off the end are lost — this can cause overflow or data loss

Exam tip: Edexcel 1CP2 typically asks you to: (1) add two 8-bit binary numbers showing working and identify any overflow, (2) perform left/right shifts and state the effect (multiply/divide by 2). Always show your carry bits in addition questions and state if overflow occurs.

⚠️ Common Mistakes

Forgetting to carry — always carry 1 when two 1s are added
Not identifying overflow — if the result needs more bits than available, overflow has occurred
Left shift causing overflow when 1s are lost off the MSB
Right shift with odd numbers — remainders are lost (e.g., 5 right-shifted = 2, not 2.5)

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Worksheet — 2.1d Binary Arithmetic

8 Edexcel-style questions · AI-marked

Q1State the four rules for binary addition (0+0, 0+1, 1+0, 1+1).[2]

✅ Mark scheme

0+0=0 [½], 0+1=1 [½], 1+0=1 [½], 1+1=0 carry 1 [½] — any correct 4 combinations for 2 marks.

Q2Calculate 00101011 + 00010110. Show your carry bits.[3]

✅ Mark scheme

43+22=65; correct carry bits shown [1]; intermediate working [1]; 01000001 [1]

Q3What is overflow and when does it occur in 8-bit arithmetic?[2]

✅ Mark scheme

Overflow is when the result is too large to be stored in the available number of bits [1]; occurs when the result exceeds 255 (unsigned) or +127/−128 (signed 8-bit) [1].

Q4Perform a left shift by 1 on 00010110. State the effect on the value.[2]

✅ Mark scheme

00101100 [1]; the value is multiplied by 2 (22 → 44) [1].

Q5Perform a right shift by 2 on 11001100. State the resulting value.[2]

✅ Mark scheme

00110011 [1]; value is 51 (divided by 4: 204÷4=51) [1].

Q6Adding 10110000 + 10100000 in 8-bit arithmetic. Does overflow occur? Explain.[2]

✅ Mark scheme

176+160=336; yes overflow occurs [1]; result (336) exceeds maximum 8-bit value (255) so a 9th carry bit is lost [1].

Q7What is the result of shifting 00000001 left by 7 places?[2]

✅ Mark scheme

10000000 [1]; value = 128 (1 × 2^7) [1].

Q8Explain why binary shifts are used instead of multiplication/division circuits in simple processors.[2]

✅ Mark scheme

Binary shifts are simpler and faster [1]; they allow multiplication or division by powers of 2 without requiring complex multiplication/division circuits [1].

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Binary Addition

Example: Add 00110101 + 00011010

Overflow

Example of overflow (8-bit):

Binary Shifts

Worksheet — 2.1d Binary Arithmetic

Mini Test — Binary Arithmetic

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