Describe the format of binary floating-point real numbers

13.3 Floating‑point Numbers, Representation and Manipulation

Floating‑point numbers let computers store real numbers (like 3.1415) in a compact binary format. Think of it as a digital version of a ruler that can stretch (float) to show numbers of any size, from tiny fractions to huge values, while keeping a fixed number of digits.

What is a floating‑point number?

A floating‑point number is written as sign × mantissa × 2^exponent.
🔢 Example: 6.5 = +1.625 × 2² (mantissa 1.625, exponent 2, sign +).

Binary floating‑point format (IEEE 754)

The standard format splits the bits into three parts:

• Sign bit – 0 for positive, 1 for negative.
• Exponent field – stores the exponent + a bias.
• Mantissa (fraction) field – stores the significant digits.

The bias is chosen so that the exponent can be stored as an unsigned integer. For single precision (32 bits): bias = 127. For double precision (64 bits): bias = 1023.

Bits	Field	Description
1	Sign	0 = +, 1 = –
8	Exponent	Exponent + bias
23	Mantissa	Fractional part (leading 1 is implicit)

Encoding an example: 6.5 in single precision

1. Convert 6.5 to binary: 110.1₂.
2. Normalise: 1.101 × 2².
3. Sign bit = 0 (positive).
4. Exponent = 2 + bias(127) = 129 → 10000001₂.
5. Mantissa = 101 followed by 20 zeros → 101000000000000000000.

Final 32‑bit pattern: 0 10000001 10100000000000000000000.
🔢 In hex: 0x40D00000.

Key concepts to remember

1. Floating‑point numbers are stored as sign × mantissa × 2^exponent.
2. The exponent is stored with a bias to allow negative exponents.
3. The mantissa has an implicit leading 1 (except for subnormal numbers).
4. Rounding occurs when the exact binary representation has more bits than the mantissa can hold.
5. Special bit patterns represent Infinity, NaN, and subnormal numbers.

Exam Tips

💡 Bias calculation: exponent field = true exponent + bias.
💡 Normalise before encoding: shift binary point so that only one non‑zero digit is left of the point.
💡 Rounding rule: round to nearest, ties to even (IEEE 754 default).
💡 Check for special cases: zero, Infinity, NaN – they have unique bit patterns.
💡 Practice converting numbers: start with decimal → binary → normalise → encode. The more you practice, the faster you’ll get.

Quick Practice Problem

Encode the decimal number −12.75 in single‑precision IEEE 754 format. Show the sign bit, exponent field, and mantissa bits.

📝 Hint: 12.75 = 1100.11₂ → normalise to 1.10011 × 2³. Then apply the bias and fill the mantissa.