Floating Point Binary Numbers

Overview

Floating point binary numbers are used to represent real numbers (numbers with fractional parts) in binary. This representation is similar to scientific notation but uses binary format. Floating-point numbers include two main parts: the mantissa and the exponent. Understanding floating-point representation and how to normalise floating-point numbers is crucial, as this affects the accuracy and range of the numbers represented in computing.

Floating Point Representation

Structure

A floating-point number is represented as:

$$\text{Mantissa} \times 2^{\text{Exponent}}$$

Mantissa and Exponent

• Mantissa: Represents the significant digits of the number, usually in two's complement for signed values.
• Exponent: Determines the position of the binary point, also in two's complement form to allow for positive and negative shifts.

Converting Denary to Floating-Point Binary

Step 1: Convert the number to binary form, separating the integer and fractional parts.

• Integer part: Use repeated division by 2.

• Fractional part: Use repeated multiplication by 2. Step 2: Determine the mantissa and exponent.

• Shift the binary point so that the mantissa starts with 1.xxxx.

• Count the shifts made to normalise the binary number; this becomes the exponent.

Two's Complement in Floating-Point

Since both the mantissa and exponent can be positive or negative, we use two's complement notation for each:

• Mantissa: Represents the signed value in normalised form.
• Exponent: Represents the power of two as a signed value.

Normalisation of Floating-Point Numbers

Purpose of Normalisation

• Normalising ensures maximum precision by adjusting the mantissa so that it starts with a significant 1.
• This prevents unnecessary leading zeros, improving accuracy.

Normalisation Rules

• For positive numbers, the mantissa should start with 01.
• For negative numbers, the mantissa should start with 10.

Examples

Note Summary