1.1.3 Binary Arithmetic and Overflow
Binary arithmetic is performed using a fixed number of bits. This matters because a result that is mathematically correct may still be impossible to store in the space available. In this section, you will work with unsigned and signed integers, use two's complement, perform binary addition and subtraction, and recognise overflow.
By the end of this section, you should be able to:
- determine the range of an unsigned or two's complement integer;
- convert between denary values and fixed-width two's complement;
- perform binary addition and subtraction;
- explain how subtraction can be carried out using two's complement;
- identify overflow in unsigned and signed calculations.
Exam focus
Always identify the representation and the number of bits before calculating. The same bit pattern can represent different values when interpreted as unsigned binary or two's complement.
Fixed-Width Integer Storage
A computer usually allocates a fixed number of bits to each integer. The available bit width determines the range of values that can be represented.
Unsigned range
An unsigned integer represents zero or a positive whole number. With n bits, its range is:
0 to 2n − 1
| Bit width | Minimum | Maximum |
|---|---|---|
| 4 bits | 0 | 15 |
| 8 bits | 0 | 255 |
| 12 bits | 0 | 4095 |
Two's complement range
With n bits, a two's complement integer has the range:
−2n−1 to 2n−1 − 1
| Bit width | Minimum | Maximum |
|---|---|---|
| 4 bits | −8 | 7 |
| 8 bits | −128 | 127 |
| 12 bits | −2048 | 2047 |
Common mistake
Do not use the unsigned range for a signed value. An 8-bit pattern can represent 0 to 255 when unsigned, but −128 to 127 when interpreted as two's complement.
Representing Negative Integers
Two's complement is a signed binary representation that allows positive and negative values to be processed using the same addition circuitry.
Two's complement: invert every bit and then add 1.
Example: represent −43 using 8 bits
| Step | Bit pattern |
|---|---|
| Write +43 using 8 bits | 00101011 |
| Invert every bit | 11010100 |
| Add 1 | 11010101 |
Therefore, 11010101 represents −43 in 8-bit two's complement.
Reading a negative two's complement value
If the most significant bit is 1, the value is negative. You can invert the bits and add 1 to find its positive magnitude, then attach a minus sign.
Alternatively, use a negative place value for the most significant bit. For an 8-bit value, the place values are:
−128, 64, 32, 16, 8, 4, 2, 1
Common mistake
Keep the required number of bits throughout the conversion. Removing leading zeros changes the bit width and may change the meaning of the result.
Interactive: Animated Two's Complement
Use this tool to follow a conversion one stage at a time. Select the bit width first, then choose the conversion direction.
Binary Addition
Add from right to left. Record one result bit in each column and carry to the next column when the total is 2 or 3.
| Column total | Write | Carry |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 0 |
| 2 | 0 | 1 |
| 3 | 1 | 1 |
Worked example
010110
+ 001101
--------
100011The result is 100011₂, which is 35₁₀.
Exam tip
Write each carry above the next column. This makes the method visible and reduces errors caused by losing a carry.
Interactive: Animated Binary Addition
This tool works from the least significant bit to the most significant bit and shows how carries affect each column.
Overflow
Overflow occurs when a result falls outside the range available for the chosen bit width and representation.
Unsigned overflow
In 4-bit unsigned binary, the largest value is 15. Therefore:
1101 (13)
+ 0101 (5)
------
1 0010The mathematical result is 18, but 18 requires five bits. Storing only the lower four bits would give an incorrect value.
Signed overflow
For two's complement addition, overflow occurs when two values with the same sign produce a result with the opposite sign.
| Operands | Result sign | Overflow? |
|---|---|---|
| positive + positive | negative | Yes |
| negative + negative | positive | Yes |
| different signs | either sign | No signed overflow |
Common mistake
A carry out of the most significant bit is not, by itself, a reliable test for signed overflow. Check the signs of the operands and the stored result.
Binary Subtraction
Direct binary subtraction uses borrowing. When 0 must subtract 1, borrow from the next non-zero column to the left.
Worked example using borrowing
101101
- 011011
--------
010010The result is 010010₂, which is 18₁₀.
Subtraction using two's complement
A computer can calculate A − B by adding A to the two's complement representation of B. This allows one addition circuit to support both operations.
Why two's complement is preferred
A computer's ALU only needs to implement addition. By representing negative numbers in two's complement, subtraction (A − B) is carried out as A + (−B), using the same adder circuit that performs addition. This avoids needing separate borrowing logic in hardware.
Common mistake
In A − B, form the two's complement of B only.
Interactive: Animated Binary Subtraction
Compare direct borrowing with the two's complement method.
Practice and Review
Try these questions
- State the range of an 8-bit unsigned integer.
- State the range of an 8-bit two's complement integer.
- Represent −43 using 8-bit two's complement.
- Convert 11101010 from 8-bit two's complement to denary.
- Add 00110110 and 00011101.
- Explain whether 01110110 + 00110101 causes signed overflow in 8 bits.
- Calculate 101101 − 011011 using binary subtraction.
- Explain why subtraction can be implemented using addition hardware.
Final checklist
- Confirm the representation.
- Keep the stated bit width.
- Show carries or borrowing.
- Check the permitted range.
- Include an overflow explanation when required.