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Revision notes for Edexcel GCSE Computer Science Unsigned and two's complement integers. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Computer Science (1CP2) specification, so the content matches what's examinable rather than general Computer Science background.

Unsigned and two's complement integers

What you'll learn

  • How 8-bit unsigned integers represent denary values from 0 to 255.
  • How two's complement represents signed integers from -128 to +127.
  • How to convert between 8-bit binary and denary for both representations.
  • How computers manipulate these integers using addition, subtraction, shifts and overflow rules.

Prerequisites: bits, bytes and place values

An integer is a whole number, such as 0, 42, -7 or 255. Computers store integers using bits, where each bit is either 0 or 1.

Definition

Bit, byte and bit pattern

A bit is a single binary digit, either 0 or 1. A byte is 8 bits. A bit pattern is a particular sequence of bits, such as 0110 1001.

For GCSE, you usually work with 8-bit binary numbers. We group the 8 bits into two nibbles of 4 bits to make them easier to read:

10110101 becomes 1011 0101

In an unsigned 8-bit number, the place values are:

Bit position76543210
Place value1286432168421

The leftmost bit is the most significant bit because it has the largest place value. The rightmost bit is the least significant bit because it has the smallest place value.

Example

Converting unsigned binary to denary

Convert 1011 0101 to denary.

  1. Match each bit to its unsigned place value: 1 0 1 1 0 1 0 1 lines up with 128 64 32 16 8 4 2 1.

  2. Add the place values where the bit is 1: 128 + 32 + 16 + 4 + 1.

  3. Calculate the total: 1011 0101 represents 181 as an unsigned integer.

Unsigned integers

An unsigned integer cannot be negative. Every bit is used for the size of the number, so 8 bits can store values from 0 to 255.

Definition

Unsigned integer

An unsigned integer is a whole number with no sign bit, so it represents zero or a positive value only.

With 8 bits, there are 28=2562^8 = 25628=256 possible bit patterns. Because counting starts at 0, the largest unsigned value is 255:

  • Smallest: 0000 0000 = 0
  • Largest: 1111 1111 = 255
Key Idea

Unsigned 8-bit range

For Edexcel GCSE, an 8-bit unsigned integer has the range 0 to 255.

Example

Representing denary as unsigned binary

Represent denary 156 as an 8-bit unsigned binary number.

  1. Start with the largest place value, 128. Since 156 is at least 128, put a 1 in the 128 column and subtract it, leaving 28.

  2. Check the next place values: 64 and 32 are too large, so put 0s there. 16 fits into 28, so put a 1 and subtract 16, leaving 12.

  3. Continue with 8, 4, 2 and 1: 8 fits, leaving 4; 4 fits, leaving 0; 2 and 1 do not fit.

  4. The bits are 1001 1100, so denary 156 is 1001 1100 as an 8-bit unsigned integer.

Signed integers and two's complement

A signed integer can be positive, negative or zero. GCSE Computer Science uses two's complement to represent signed integers.

The important idea is that the same 8-bit pattern can mean different denary values depending on whether it is being interpreted as unsigned or two's complement.

Comparison of 8-bit unsigned and two's-complement signed integer ranges

Definition

Two's complement

Two's complement is a method for representing signed binary integers where, in 8 bits, the leftmost bit has the value -128 instead of +128.

For 8-bit two's complement, the place values are:

Bit position76543210
Place value-1286432168421

This gives the range:

  • Smallest: 1000 0000 = -128
  • Largest: 0111 1111 = +127
Tip

Check the sign bit

In 8-bit two's complement, if the leftmost bit is 0, the value is zero or positive. If the leftmost bit is 1, the value is negative.

Example

Interpreting a two's-complement number

Convert 1101 0110 from 8-bit two's complement to denary.

  1. Use the two's-complement place values: -128 64 32 16 8 4 2 1.

  2. Add the values where the bit is 1: -128 + 64 + 16 + 4 + 2.

  3. Calculate the result: -128 + 64 + 16 + 4 + 2 = -42, so 1101 0110 represents -42.

Converting a negative denary number to two's complement

To represent a negative number in 8-bit two's complement, one reliable method is:

  1. Write the positive version in 8-bit binary.
  2. Flip every bit: 0 becomes 1, and 1 becomes 0.
  3. Add 1.

This is called the “flip and add 1” method.

Example

Representing a negative number in two's complement

Represent -37 as an 8-bit two's-complement number.

  1. Write +37 in 8-bit binary: 37 = 0010 0101.

  2. Flip every bit: 0010 0101 becomes 1101 1010.

  3. Add 1: 1101 1010 + 0000 0001 = 1101 1011.

  4. Therefore, -37 is represented as 1101 1011.

Common Mistake

Forgetting the fixed 8-bit width

Always keep all 8 bits when using two's complement. For example, +5 is 0000 0101, not just 101, before you flip bits.

Common Mistake

The edge of the range

In 8-bit two's complement, -128 is valid but +128 is not. The range is -128 to +127, so a result of +128 causes overflow.

Manipulating integers: addition and overflow

Computers manipulate integers by carrying out binary operations such as addition. The bit addition rules are the same whether the number is unsigned or two's complement, but the meaning of the result depends on how you interpret the bits.

Definition

Overflow

Overflow happens when the result of a calculation is outside the range that can be represented using the available number of bits.

For 8-bit unsigned integers, overflow happens if the result is less than 0 or greater than 255.

For 8-bit two's-complement integers, overflow happens if the result is less than -128 or greater than +127.

8-bit binary addition examples showing how overflow depends on unsigned or two's-complement interpretation

Example

Adding two signed integers

Add 75 and 90 using 8-bit two's complement.

  1. Convert both positive numbers to 8-bit binary: 75 is 0100 1011 and 90 is 0101 1010.

  2. Add the bit patterns: 0100 1011 + 0101 1010 = 1010 0101.

  3. Interpret the result as two's complement. The leftmost bit is 1, so the result is negative: -128 + 32 + 4 + 1 = -91.

  4. Since adding two positive numbers should not give a negative number, and the true answer 165 is outside the range -128 to +127, overflow has occurred.

Tip

Signed overflow sanity check

In two's complement, adding two positive numbers should give a positive result. Adding two negative numbers should give a negative result. If the sign unexpectedly changes, suspect overflow.

Subtraction by adding a negative

Two's complement is useful because subtraction can be handled as addition of a negative number.

For example:

6 - 9 can be treated as 6 + (-9)

Example

Subtracting using two's complement

Calculate 6 - 9 using 8-bit two's complement.

  1. Rewrite the subtraction as addition: 6 + (-9).

  2. Represent both numbers in 8-bit two's complement: 6 is 0000 0110; -9 is found from 256 - 9 = 247, so it is 1111 0111.

  3. Add the bit patterns: 0000 0110 + 1111 0111 = 1111 1101.

  4. Interpret 1111 1101 as two's complement: -128 + 64 + 32 + 16 + 8 + 4 + 1 = -3, so the calculation gives -3.

Binary shifts

A binary shift moves all bits left or right. Shifts are a quick way to multiply or divide by powers of 2, but bits can be lost, so overflow and sign changes matter.

A logical shift fills the empty positions with 0s. It is normally used with unsigned integers.

An arithmetic shift preserves the sign of a two's-complement signed integer. For an arithmetic right shift, the leftmost bit is copied into the new empty spaces.

Key Idea

What shifts do

A left shift by one place usually multiplies by 2. A right shift by one place usually divides by 2 using integer division, but signed negative numbers need an arithmetic right shift to keep the sign correct.

Example

Arithmetic right shift of a negative number

Arithmetic right shift 1111 1000 one place to the right.

  1. Interpret the original number as two's complement: 1111 1000 has value -128 + 64 + 32 + 16 + 8 = -8.

  2. Shift all bits one place to the right and copy the sign bit 1 into the leftmost position: 1111 1000 becomes 1111 1100.

  3. Interpret the result: 1111 1100 has value -128 + 64 + 32 + 16 + 8 + 4 = -4, so the arithmetic right shift has divided -8 by 2.

Common Mistake

Using a logical shift on a signed negative number

A logical right shift of 1111 1000 would give 0111 1100, which is positive. For two's-complement negative numbers, use an arithmetic right shift to preserve the sign.

Exam technique

In the exam

  1. First identify the representation: unsigned or two's complement. The same bit pattern can mean different denary values.

  2. Write the 8-bit place values above the bits before converting. Use 128 64 32 16 8 4 2 1 for unsigned, but -128 64 32 16 8 4 2 1 for two's complement.

  3. Check the allowed range: unsigned is 0 to 255; two's complement is -128 to +127.

  4. For addition, calculate the 8-bit result, then decide whether overflow has occurred based on the chosen representation.

  5. For right shifts, decide whether the question needs a logical shift or an arithmetic shift.

Self review

Check yourself

  • What denary value does 1111 1111 represent as unsigned, and what does it represent as two's complement?
  • How would you represent -18 using 8-bit two's complement?
  • Why does 0111 1111 + 0000 0001 cause overflow for two's complement but not for unsigned?
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