Sign, Exponent, and Mantissa Explained

Every IEEE 754 floating-point number is composed of three fields that work together to represent a value using scientific notation in base 2. Understanding each field is the key to mastering floating-point arithmetic.

1. The Sign Bit

The sign bit is the most significant bit. It is 0 for positive numbers (including +0 and +Infinity) and 1 for negative numbers (including -0 and -Infinity). For NaN values, the sign bit can be either 0 or 1.

2. The Exponent Field (biased)

The exponent is stored with a bias so that negative exponents can be represented without a separate sign bit. For float32, the bias is 127; for float64, it is 1023. The actual exponent equals the stored value minus the bias.

Special exponent values:

• All zeros (0): denormalized number or zero
• All ones (255 for float32, 2047 for float64): infinity or NaN
• 1 to 254 (float32) / 1 to 2046 (float64): normalized number

3. The Mantissa (Significand)

For normalized numbers, there is an implicit leading 1 that is not stored. The mantissa field represents the fractional part. So the actual significand is 1.mantissa_bits in binary.

For denormalized numbers, the implicit bit is 0 instead of 1, giving a significand of 0.mantissa_bits.

Worked example: -6.5

1. Convert 6.5 to binary: 110.1
2. Normalize: 1.101 x 2^2
3. Sign = 1 (negative)
4. Exponent = 2 + 127 = 129 = 10000001 in binary
5. Mantissa = 101 followed by 20 zeros (23 bits total)

Result: 1 10000001 10100000000000000000000 = 0xC0D00000

Verification: (-1)^1 x (1 + 0.5 + 0.125) x 2^2 = -1 x 1.625 x 4 = -6.5

The implicit bit trick:

By not storing the leading 1 for normalized numbers, IEEE 754 gains one extra bit of precision for free. This means float32 effectively has 24 bits of significand precision even though only 23 are stored, and float64 has 53 bits even though only 52 are stored.