Floating Point Numbers

Representing fractions in binary

To store numbers with fractional parts (and very large/small numbers), computers use floating point representation — the binary equivalent of standard form. A floating point number is split into two parts:

number = mantissa × 2^exponent

• The mantissa (or significand) holds the significant digits.
• The exponent says where the binary point goes (how far to shift it).

Both are stored in a fixed number of bits (e.g. an 8-bit mantissa and a 4-bit exponent), usually in two's complement so both can be negative.

Normalisation

A floating point number is normalised to use the available bits most precisely and to give a unique representation. For a two's complement mantissa:

• Positive numbers start 0.1… (a 0 sign bit followed by a 1).
• Negative numbers start 1.0… (a 1 sign bit followed by a 0).

Normalising means shifting the mantissa so the first two bits differ, and adjusting the exponent to compensate. This maximises the number of significant figures stored.

The precision–range trade-off

For a fixed total number of bits:

• More mantissa bits → greater precision (accuracy of the value).
• More exponent bits → greater range (how large or small a number can be).
• Increasing one reduces the other — a key design trade-off.

Errors

• Rounding/representation errors: many decimals (e.g. 0.1) cannot be stored exactly in binary, causing small inaccuracies.
• Absolute error = the actual difference; relative error = error as a proportion of the true value.
• Overflow (number too large for the exponent) and underflow (too small) can also occur.

Fixed vs floating point

• Fixed point places the binary point at a set position — simpler and faster, but limited range.
• Floating point moves the point (via the exponent) — much larger range and better for scientific values, but more complex and can be less precise.

Worked example

Why is the mantissa normalised before storing?

• Normalising shifts out leading redundant bits so the mantissa carries the maximum number of significant figures, giving the greatest precision for the bits available (and a unique representation). ✓

Common mistakes

• Mixing up the roles: mantissa = significant digits, exponent = position of the point.
• Forgetting normalised two's complement starts 0.1 (positive) or 1.0 (negative).
• Thinking more bits fixes everything — there's always a precision vs range trade-off.

Exam tips

• Be able to state what the mantissa and exponent represent and write number = mantissa × 2^exponent.
• Explain normalisation in terms of maximising precision.
• Discuss the precision–range trade-off and sources of rounding error.

Key facts to remember

• Floating point: mantissa × 2^exponent; mantissa = significant figures, exponent = position of the binary point (often two's complement).
• Normalised two's complement mantissa starts 0.1 (positive) / 1.0 (negative) → maximum precision.
• Fixed bits give a precision (mantissa) vs range (exponent) trade-off; not all decimals store exactly → rounding errors.