OCR · GCSE · Computer Science
Course version: J277 · Version from Sept 2022
How to add two binary integers together (up to and including 8 bits) and…
Full specification wording
Data storage: How to add two binary integers together (up to and including 8 bits) and explain overflow errors which may occur
Binary addition operates column by column from the least significant bit (rightmost), similar to denary addition but using only 0 and 1.
Checked against the OCR J277 specification on .
- Point 2 of 13
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Binary addition operates column by column from the least significant bit (rightmost), similar to denary addition but using only 0 and 1. The single-bit rules are: 0+0=0, 0+1=1, 1+0=1, 1+1=10₂ (write 0, carry 1), and 1+1+1=11₂ (write 1, carry 1). For example, adding 0110 1101₂ and 0011 0111₂: starting at the right, 1+1=10₂ (write 0, carry 1); next 0+1+1=10₂ (write 0, carry 1); then 1+1+1=11₂ (write 1, carry 1); continuing leftwards yields 1010 0100₂. In an 8-bit system, the maximum representable value is 255. If an addition exceeds this (e.g., 200+100=300), a carry is generated out of the most significant bit (the 128s column). Because there is no ninth bit to store this carry, it is discarded, causing an overflow error. The stored result is incorrect, specifically 256 less than the true sum (e.g., 300 becomes 44).
What earns the marks
- State the single-bit addition rules, explicitly showing that 1+1=10₂ and 1+1+1=11₂.
- Add two 8-bit binary numbers column by column from the least significant bit, correctly processing carries.
- Define an overflow error as occurring when a carry is generated out of the most significant bit (eighth bit) and cannot be stored.
- Explain that an overflow error results in an inaccurate answer because the discarded ninth bit has a place value of 256.
Common mistakes and how to put them right
Adding from left to right.
Put it right: Start at the least significant bit (rightmost) and move left.
Writing '2' in a binary column.
Put it right: Write 0 and carry 1 to the next column, as 1+1=10₂.
Ignoring the final carry out of the 8th bit.
Put it right: Explicitly state that the carry is discarded, resulting in an overflow error.
Exam tips
- Align the two 8-bit numbers vertically and write any carries in a clear row above or below the main columns.
- If a calculation produces a ninth bit, explicitly state that an overflow error has occurred because the register is limited to 8 bits.
Try the lesson
The start of the Data storage: How to add two binary integers together (up to and including 8 bits) and explain overflow errors which may occur lesson, made from checked teaching notes for this specification point.
Have a go first
A computer adds the unsigned 8-bit binary integers 01101100 and 00110101. What is the 8-bit result and does an overflow error occur?
Listen
Binary addition operates column by column from the least significant bit (rightmost), similar to denary addition but using only 0 and 1. The single-bit rules are: 0+0=0, 0+1=1, 1+0=1, 1+1=10₂ (write 0, carry 1), and 1+1+1=11₂ (write 1, carry 1). For example, adding 0110 1101₂ and 0011 0111₂: starting at the right, 1+1=10₂ (write 0, carry 1); next 0+1+1=10₂ (write 0, carry 1); then 1+1+1=11₂ (write 1, carry 1); continuing leftwards yields 1010 0100₂. In an 8-bit system, the maximum representable value is 255. If an addition exceeds this (e.g., 200+100=300), a carry is generated out of the most significant bit (the 128s column). Because there is no ninth bit to store this carry, it is discarded, causing an overflow error. The stored result is incorrect, specifically 256 less than the true sum (e.g., 300 becomes 44). A strong answer earns marks for these points. State the single-bit addition rules, explicitly showing that 1+1=10₂ and 1+1+1=11₂. Add two 8-bit binary numbers column by column from the least significant bit, correctly processing carries. Define an overflow error as occurring when a carry is generated out of the most significant bit (eighth bit) and cannot be stored. Explain that an overflow error results in an inaccurate answer because the discarded ninth bit has a place value of 256. A common slip: Adding from left to right: this causes carries to move in the wrong direction. Instead, start at the least significant bit (rightmost) and move left. Exam tip: Align the two 8-bit numbers vertically and write any carries in a clear row above or below the main columns.
Key words
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Single-bit rules
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Least significant bit
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Overflow error
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Discarded carry
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See every definition
- Single-bit rules
- The rules for adding one bit to one bit: 0+0=0, 0+1=1, 1+0=1, 1+1=10₂ (write 0, carry 1), and 1+1+1=11₂ (write 1, carry 1).
- Least significant bit
- The rightmost bit, where column-by-column binary addition starts, moving leftwards while processing carries.
- Overflow error
- Occurs when a carry is generated out of the most significant bit (the 128s column) and, with no ninth bit to store it, is discarded.
- Discarded carry
- The carry out of the eighth bit has a place value of 256, so discarding it makes the stored result incorrect, specifically 256 less than the true sum.
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