13.3 Floating-point numbers, representation and manipulation

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Floating-point numbers: why they are needed

A real number is a number that can have a fractional part, such as 4.5, 0.25, or -10.375.

A computer can store numbers in different ways. One way is fixed-point. In fixed-point, the binary point stays in one fixed position. This is simple, but it has two big limits:

• the range of values is limited
• it is not good at storing both very large and very small numbers

To solve this, computers can use floating-point representation. In floating-point, the binary point is allowed to “move”. This makes it possible to store a much wider range of values.

Floating-point works in a similar way to standard form in maths.

For example, in denary:

• 3450 can be written as 3.45 × 10³

In binary floating-point:

• a number is written in the form M × 2ᴱ

Here:

• M is the mantissa
• E is the exponent

So the mantissa gives the main digits of the number, and the exponent tells you how far the binary point moves.

The format of a binary floating-point number

A binary floating-point number is split into two parts:

• the mantissa
• the exponent

The general form is:

value = mantissa × 2^exponent

In this syllabus, floating-point values use two’s complement form. This means signed values can be stored, including negative values.

A common example is:

• 8 bits for the mantissa
• 8 bits for the exponent

That gives a total of 16 bits.

The mantissa is treated as a signed binary fraction. The binary point is placed after the first bit.

So if the mantissa is:

01011010

it is read as:

0.1011010

If the mantissa is negative, it is still stored using two’s complement.

The exponent is a signed integer stored in two’s complement as well.

So the whole floating-point value is:

mantissa × 2^exponent