Normalise floating-point numbers

13.3 Floating‑point numbers, representation and manipulation

What is a floating‑point number?

A floating‑point number is like a decimal number that can grow or shrink its “decimal point” (the exponent) so that we can represent very large or very small values with a fixed number of bits. Think of it as a phone number that can be written in a short form: +1 (555) 123‑4567 → +15551234567 – the format stays the same, but the value is the same.

IEEE‑754 format (simplified)

Most computers use the IEEE‑754 standard. A 32‑bit single‑precision number is split into three parts:

Why do we need normalisation?

Normalisation ensures that the mantissa starts with a leading 1 (for normal numbers). This gives us the most precision for the value we can store. If we didn’t normalise, we could lose significant digits.

Normalisation process (step‑by‑step)

1. Write the number in binary scientific notation. Example: $-13.375_{10} = -1101.011_{2}$
2. Shift the binary point so that there is exactly one non‑zero digit to the left. $-1101.011_{2} = -1.101011 \times 2^{3}$
3. Record the sign bit. Sign = 1 (because the number is negative)
4. Calculate the exponent. Exponent = 3 (from the shift) Stored exponent = $3 + 127 = 130$ → binary $10000010$
5. Store the mantissa. Drop the leading 1 (it’s implicit) and keep the next 23 bits: $10101100000000000000000$
6. Combine the parts. Final 32‑bit pattern: 1 10000010 10101100000000000000000

Illustrative example

Let’s normalise $0.00015625_{10}$.

• Binary: $0.00015625 = 0.000001010000_{2}$
• Shift to get one 1 left of the point: $1.010000 \times 2^{-4}$
• Sign bit: 0 (positive)
• Exponent: $-4 + 127 = 123$ → binary $01111011$
• Mantissa: drop leading 1 → $01000000000000000000000$
• Resulting 32‑bit pattern: 0 01111011 01000000000000000000000

Common pitfalls

• Underflow. Numbers too small to be normalised become denormalised or zero.
• Overflow. Numbers too large cannot be represented and become infinity.
• Rounding errors. When the mantissa is truncated, the value may be slightly off.

Quick practice

Try normalising the following numbers. Write down the sign bit, exponent, and mantissa.

1. $5.25_{10}$
2. $-0.75_{10}$
3. $1024_{10}$

Take‑away

Normalising a floating‑point number is like adjusting a camera lens: you zoom in until the main subject (the leading 1) is clear, then record the zoom level (the exponent) and the details (the mantissa). This keeps the picture sharp across a huge range of values.