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OCR H446 1.4.1 Binary integer arithmetic
Part 4 of 11 · H446 1.4.1 · Data types
Fixed-width addition and subtraction is where carries, borrows and overflow become examinable in H446 1.4.1, and the working carries marks as much as the answer does. A game server combining score changes gives the context for 8-bit addition, borrowing subtraction and two's-complement subtraction, before separating a carry out from signed overflow.
Students will:
- add pairs of 8-bit unsigned values and record the carry row
- subtract 8-bit values by borrowing and by adding a two's complement
- hold a result at the stated width and interpret the remaining bits
- decide whether an addition has overflowed and give the representable-range reason
- explain in writing why a carry out and signed overflow are not the same event
Inside: 8 explanation cells, 1 fill-in-the-blanks cell, 1 written answer and 4 number answers. 37 marks, about 45 to 55 minutes.
Series: H446 1.4.1 · Data types, part 4 of 11.
Shared by Coding PathwayVerified teacher
- 14 cells
- About 45 minutes
- CC BY-SA 4.0
- Shared 31 Aug 2026
- Updated 1 Oct 2026
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The whole resource, exactly as a class sees it. Answers and marking are held back.
Binary integer arithmetic
A game server combines score changes and corrects counters. Fixed width makes the working and any unrepresentable result visible.
By the end, you will be able to
- add binary values with carries
- subtract using borrowing or two’s-complement addition
- distinguish unsigned carry from signed overflow
- interpret whether a stored result is meaningful
Reactivate: 1 + 1 in binary produces result bit 0 and carry 1.
Addition
Work from right to left. In one column: 0 + 0 = 0; 0 + 1 = 1; 1 + 1 = 10; and 1 + 1 + carry 1 = 11. Write the result bit and carry the other bit to the next column.
Worked example: 00101101 (45) + 00010111 (23) = 01000100 (68). Work through all eight columns and retain each carry above the next column; the denary values are a check, not a replacement for the binary method.
Worked model: align every column
carry: 1 1 1 1
00110110
+ 00011101
----------
01010011
Work right to left. The denary check is 54 + 29 = 83, and 01010011₂ is 83. Use denary only as a check; keep binary working for the assessed method.
Add each pair of 8-bit unsigned binary values. Show and mark the carry row.
a)Add these 8-bit binary numbers: 10100011 + 00111011.
b)Add these 8-bit binary numbers: 01011011 + 01111010.
c)Add these 8-bit binary numbers: 01100110 + 01111111.
d)Add these 8-bit binary numbers: 01011010 + 10011101.
Subtraction
You may use binary borrowing for unsigned values. For signed two's-complement subtraction, form the two's complement of the subtrahend and add it. Keep the width fixed and interpret the final pattern.
Worked signed example: 9 - 6 becomes 00001001 + 11111010, because 11111010 is -6 in 8-bit two's complement. The fixed-width sum is 00000011, so the result is +3. Discarding a carry beyond the eighth bit is part of keeping the stated width; it is not automatically signed overflow.
Model two’s-complement subtraction
To calculate A − B at a fixed width: form the two’s complement of B, add it to A, discard a carry beyond the fixed width, then interpret the remaining bits in the stated signed system.
Guided checkpoint: before accepting the bits, estimate the sign and range of the mathematical answer. If two positive inputs in signed addition produce a negative-looking result, investigate overflow rather than reporting it blindly.
Subtract the second 8-bit unsigned value from the first. Show the borrowing/carry working.
a)Work out 11100101 − 00111011 in binary.
b)Work out 01010011 − 01001101 in binary.
c)Work out 00100110 − 00001110 in binary.
d)Work out 10110010 − 01011001 in binary.
Complete each 8-bit two's-complement subtraction and record the method.
a)Use two's complement to work out 01010000 − 01001001.
b)Use two's complement to work out 01011010 − 00000010.
c)Use two's complement to work out 01111011 − 01101111.
Carry and overflow are not synonyms
A carry out shows an unsigned result needs more bits. Signed overflow occurs when a two's-complement result is outside the representable range; for addition, two inputs with the same sign producing a result with the opposite sign is a warning.
Compare two 8-bit signed cases:
- 11111111 (-1) + 11111111 (-1) gives stored result 11111110 (-2) with a carry out. The signed result is valid, so there is no signed overflow.
- 01000000 (+64) + 01000000 (+64) gives 10000000. The mathematical answer +128 is outside -128 to +127, so signed overflow has occurred even though a carry-out rule alone would not diagnose it.
Always test the interpretation and representable range named in the question.
For each 8-bit addition, decide whether overflow occurs and explain the representable-range reason.
a)Add 00010000 and 00000011 in an 8-bit register, then say what happens to the answer.
b)Add 00100111 and 00011000 in an 8-bit register, then say what happens to the answer.
c)Add 00110010 and 10111101 in an 8-bit register, then say what happens to the answer.
d)Add 10011110 and 10100011 in an 8-bit register, then say what happens to the answer.
Two positive 8-bit two's-complement values are added and the result begins with 1. Explain what has happened and why keeping only the 8-bit result would be misleading.
Refer to sign, representable range and the interpretation of the stored result.
Students type their answer here.
Closed-book checkpoint
Complete each sentence from memory. There is no answer bank and correctness is held for teacher review.
Retrieval check
Explain the difference between a carry out and signed overflow, and give the four stages for fixed-width two’s-complement subtraction.