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1CP2-P-1.4 Signed integers and two's complement foundations
Part 4 of 5 · 1CP2-P-1 · Binary foundations
The signed representation worksheet, introducing two's complement as the method behind the negative patterns.
Students will:
- decide whether a value needs a signed or an unsigned representation
- state the 8-bit ranges for both interpretations
- interpret an 8-bit two's complement pattern in denary
- represent a negative denary value using the full invert-and-add method
- diagnose the recurring error of stopping after the inversion
Inside: 9 explanation cells, 1 multiple-choice question, 4 number cells, 3 fill-in-the-blanks cells and 1 written answer. 20 marks, about 45 minutes.
Series: 1CP2-P-1 · Binary foundations, part 4 of 5.
Shared by Coding PathwayVerified teacher
- 18 cells
- About 45 minutes
- CC BY-SA 4.0
- Shared 17 Aug 2026
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The whole resource, exactly as a class sees it. Answers and marking are held back.
Signed integers and two's complement foundations
Unsigned binary represents zero and positive values. Some contexts also require negative whole numbers. This worksheet introduces 8-bit two's complement and gives you one checked method for representing and interpreting negative values.
1. Signed or unsigned?
An unsigned integer has no negative values, so all patterns can represent zero or a positive magnitude. A signed integer can represent negative and positive whole numbers.
Choose from the context:
| Data | Suitable choice | Reason |
|---|---|---|
| Number of books on a shelf | Unsigned | A count cannot be below zero. |
| Temperature relative to 0 °C | Signed | Values may be below or above zero. |
| Change in an account balance | Signed | A change may be negative or positive. |
- magnitude
- range
- sign
- signed
- unsigned
2. The 8-bit ranges
A fixed width gives the same 256 patterns but divides them differently:
| Interpretation | Range |
|---|---|
| 8-bit unsigned | 0 to 255 |
| 8-bit two's complement signed | −128 to +127 |
Do not mix the two interpretations. The pattern 11111111 means 255 when unsigned but −1 in two's complement.
- 127
- 128
- 255
- +127
- +255
3. Interpret a negative pattern
In 8-bit two's complement, the most-significant bit has place value −128. The remaining headings stay positive.
For a negative pattern, add every selected place value, including the negative −128. This is not sign and magnitude: the remaining seven bits are not simply the magnitude of the negative value.
Interpret these 8-bit two's complement patterns in denary.
Use the signed place values −128, 64, 32, 16, 8, 4, 2, 1.
a)What is the denary value of the two's complement number 11101100?
b)What is the denary value of the two's complement number 10110110?
Explain the role of the MSB
A pattern beginning with 1 is negative only when the interpretation is two's complement. In that context, the leading bit selects −128. In an unsigned context, the same leading bit selects +128.
Always state the representation before assigning a value.
Explain why the pattern 10000001 can represent 129 in one context but −127 in another.
Name both interpretations and the different value of the most-significant place.
Students type their answer here.
4. Represent a negative value
To represent a negative denary value in 8-bit two's complement:
- write the positive magnitude in exactly eight bits;
- invert every bit, changing 0 to 1 and 1 to 0;
- add binary 1, keeping eight result bits;
- cross-check with signed place values.
The +1 is essential. Inverting alone creates one's complement, not the required two's complement.
- 0
- 1
- invert
- magnitude
- remove
- sign
Complete the full method for −18.
First write positive 18 in eight bits. Invert every bit. Then add binary 1 to produce the final two's complement pattern.
Write −18 as an 8-bit two's complement number.
Write each denary value as an 8-bit two's complement pattern.
Choose a reliable method, but enter only each final eight-bit pattern.
a)Write −73 as an 8-bit two's complement number.
b)Write −100 as an 8-bit two's complement number.
6. Read negative patterns independently
Choose either reliable interpretation method:
- add the selected signed place values; or
- invert and add 1 to recover the positive magnitude, then make the denary value negative.
If you use the second method to interpret, remember that the operation finds the magnitude; the original pattern represents a negative number.
Give the denary values of these 8-bit two's complement patterns.
Use either signed place values or recover the magnitude with invert and add one.
a)What is the denary value of the two's complement number 11010110?
b)What is the denary value of the two's complement number 10011011?
7. Diagnose recurring errors
Reject these methods:
- Sign magnitude: reading the leading 1 only as a minus sign. This representation is not required for 1CP2.
- Invert and stop: the missing add-one step produces the wrong neighbouring value.
- Drop leading bits: the task requires a complete 8-bit pattern.
- Ignore context: a leading 1 does not make an unsigned value negative.
A useful check evaluates the final pattern using −128 64 32 16 8 4 2 1.
A student writes +37 as 00100101, inverts it to 11011010 and stops. What must they do next to obtain −37 in two's complement?
- ARemove the leading 1.
- BAdd binary 1 to obtain 11011011.
- CInvert only the rightmost four bits again.
- DTreat 11011010 as sign and magnitude.
Foundations established
You can choose signed or unsigned representation, state both 8-bit ranges, interpret the negative MSB and use invert-and-add-one to construct negative two's-complement patterns.
The checkpoint retrieves this foundation with low weighting. 1CP2-P-2.1 will return to signed conversion with less scaffolding, mixed directions and range reasoning before arithmetic shifts.