Floating-Point Basics (Cambridge (CIE) A Level Computer Science): Revision Note

Floating-point binary

What is floating-point binary?

• Floating-point binary is a method of representing real numbers in binary

• Including fractions and very large or small values

• By using scientific notation in base-2

• Unlike fixed-point binary (which has a fixed position for the binary point)

  • Floating-point binary allows the point to “float” by using a mantissa and an exponent

• This allows:

  • Representation of very large and very small numbers

  • More precision for fractional values

  • Efficient use of limited storage (e.g. 8-bit, 16-bit)

Components of floating-point

• For example, in decimal:

3.14 × 10³
Mantissa = 3.14
Exponent = 3

• In binary, floating point works the same way, but uses powers of 2:

1.11 × 2⁴ → Mantissa: 1.11, Exponent: 4

Representation in binary

• A floating-point binary number typically includes:

  • A sign bit

  • A mantissa (binary digits with a fixed or implied binary point)

  • An exponent (with a sign, often using two’s complement)

• Example using 8 bits:

  0       1001       0110  
[Sign] [Exponent] [Mantissa]

Positive floating-point representation

• The MSB (most significant bit) is 0 → positive number

• The exponent shifts the binary point right or left

• Allows precision with fractional values

• Benefit: Increased precision

• Limitation: Lower maximum value compared to integer binary

Negative floating-point representation

• Two’s complement is used for negative values

• The MSB of the exponent or mantissa is used to indicate sign

• The binary point is shifted using the (possibly negative) exponent

Example: Representing 6.25 in binary

• Step 1: Convert the integer part (6) to binary

6 in binary = 110

• Step 2: Convert the fractional part (0.25) to binary

  • Multiply by 2, keep the integer part, and repeat with the remainder:

0.25 × 2 = 0.50 → 0
0.50 × 2 = 1.00 → 1

• So

0.25 in binary = 01

• Step 3: Combine integer and fractional parts

6.25 in binary = 110.01

Normalising floating-point numbers

• A floating point binary number is said to be normalised when the mantissa begins with 01 (for positive numbers) or 10 (for negative numbers)

• This format ensures:

  • Maximum precision from the available bits

  • A consistent standard for binary arithmetic and comparisons

Why normalise?

Steps to normalise a floating-point number

1. Shift the binary point until the mantissa starts with 01 (positive) or 10 (negative)

2. Adjust the exponent each time you move the binary point:

   • Moving the point left ➝ increase the exponent

   • Moving the point right ➝ decrease the exponent

Example

• Before normalisation

  • Mantissa: 0.0011

  • Exponent: 0010 (which is +2 in binary)

• Process

  • Shift binary point 2 places right to make mantissa start with 01

    • Mantissa becomes: 0.1100

    • Exponent is reduced by 2 → becomes: 0000 (which is 0 in binary)

• After normalisation

  • Mantissa: 0.1100

  • Exponent: 0000