Representing positive and negative integers using two's complement, and representing real numbers using floating-point with a mantissa and exponent.

What this key area is asking

The SQA wants you to know how a computer represents numbers in binary: positive and negative integers using two's complement, and real numbers using floating-point with a mantissa and an exponent. You should be able to convert a small value to two's complement and explain the floating-point trade-off.

Why binary, and why two's complement

A computer's memory holds only two states, so all numbers are stored in binary (base 2). Positive whole numbers are straightforward binary. The challenge is negative numbers, and the standard solution is two's complement.

Two's complement is used because it lets the processor subtract by adding the negative: the same adder circuit handles both addition and subtraction, and there is only one representation of zero. This is simpler and cheaper in hardware than the alternatives.

Working in two's complement

In $n$ bits, two's complement represents values from $-2^{n-1}$ to $+2^{n-1}-1$. In 8 bits that is $-128$ to $+127$.

For example $+5 = 0000\,0101$; inverting gives $1111\,1010$; adding 1 gives $1111\,1011 = -5$. The leading 1 confirms it is negative.

Floating-point representation of real numbers

Whole numbers are not enough: programs need real numbers such as 3.14, and very large or very small magnitudes. These use floating-point representation, the binary version of scientific notation.

This is why it is called floating-point: the same bits can represent a tiny fraction or a huge number by changing the exponent, just as $6.02 \times 10^{23}$ and $1.6 \times 10^{-19}$ share a format in decimal.

The precision versus range trade-off

The total number of bits for a floating-point number is fixed, and it is split between the mantissa and the exponent.

So a system that needs to store extremely large and small numbers favours the exponent (range); a system that needs many accurate significant figures favours the mantissa (precision). Designers choose the split for the job.

Examples in context

These representations underpin real computing. Two's complement is the integer format in virtually every processor, which is why an 8-bit signed value wraps from $+127$ to $-128$ - a real cause of bugs when a counter overflows. Floating-point (the IEEE 754 standard) is how spreadsheets, games and scientific software store decimals, and the precision limit explains why $0.1 + 0.2$ does not give exactly $0.3$ in many languages: 0.1 has no exact finite binary mantissa. Understanding range versus precision explains why scientific code sometimes uses "double precision" (more bits) to reduce rounding error.

Try this

Q1. State the two steps used to negate a number in two's complement. [2 marks]

• Cue. Invert all the bits, then add 1.

Q2. State which part of a floating-point number determines its precision. [1 mark]

• Cue. The mantissa.

Q3. For a fixed number of bits, state what increasing the exponent bits does. [1 mark]

• Cue. It increases the range (at the cost of precision).