Floating Point Addition & Subtraction

Overview

Floating-point arithmetic is crucial for handling real numbers in computing, allowing for both very large and very small values. However, adding and subtracting floating-point numbers involves extra steps compared to integers due to the mantissa and exponent structure. It's important to understand how to correctly align and normalise floating-point numbers during arithmetic operations, especially when dealing with both positive and negative values.

Structure of Floating-Point Numbers

Mantissa: Represents the significant digits of the number.
Exponent: Determines the position of the binary point, effectively scaling the number.
Sign Bit:
- 0 indicates a positive number.
- 1 indicates a negative number (using two's complement for the mantissa).

Steps for Floating-Point Addition and Subtraction

To perform addition or subtraction with floating-point numbers:

Align the Exponents:

If the exponents differ, shift the mantissa of the smaller exponent to the right until the exponents match.

Add or Subtract the Mantissas:

Perform the addition or subtraction on the aligned mantissas.

Normalise the Result:

Adjust the mantissa and exponent to ensure the mantissa is normalised (starts with 01 for positive or 10 for negative values).

Handle Overflow or Underflow:

Ensure the result fits within the given bit constraints for mantissa and exponent.

Examples

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Example 1: Addition of Two Positive Floating-Point Numbers Add $1.101 \times 2^3$ and $1.001 \times 2^1$

Step 1: Align Exponents:

Larger exponent: $2^3$

Shift the mantissa of $1.001 \times 2^1$ to match $2^3$ :

1.001 \times 2^1 = 0.01001 \times 2^3

Step 2: Add Mantissas:

1.101 + 0.01001 = 1.11101

Step 3: Normalise:

The result is already normalised: $1.11101 \times 2^3$

Final Answer:

1.11101 \times 2^3

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Example 2: Subtraction of a Positive and Negative Number Subtract $1.101 \times 2^4$ from $-1.010 \times 2^3$

Step 1: Align Exponents:

Larger exponent: $2^4$

Shift the mantissa of $-1.010 \times 2^3$

-1.010 \times 2^3 = -0.1010 \times 2^4

Step 2: Subtract Mantissas:

1.101 - (-0.1010) = 1.101 + 0.1010 = 10.010

Step 3: Normalise:

10.010 \times 2^4 = 1.001 \times 2^5

Final Answer:

1.001 \times 2^5

Note Summary

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Common Mistakes

Misaligning Exponents: Ensure the mantissas are adjusted correctly to match the exponents before adding or subtracting.
Ignoring Sign Bits: For negative numbers, remember to handle signs correctly (e.g., using two's complement).
Skipping Normalisation: Always normalise the result to maintain proper floating-point representation.

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Key Takeaways

Floating-Point Arithmetic: Involves aligning exponents, performing arithmetic on mantissas, and normalising results.
Positive & Negative Numbers: Be cautious with sign handling during subtraction and addition.
Normalisation: Ensures precision and correct representation after calculations.

Floating Point Addition & Subtraction Simplified Revision Notes for A-Level OCR Computer Science