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Floating Point Representation • Number Representation Systems • Base Conversions • Binary Arithmetic • Micro-Operations • Arithmetic Logic Units (ALUs) • The 74181 ALU • SIMD Extensions • Key Takeaways

Floating Point Representation

The Implied 1

In floating point representations (float, double, long double), there is an implied 1 that we don't actually store.

Why? Because floating point numbers are normalized.

Normalization Process

When we represent a floating point number, we normalize it to the form:

1.fraction × base^exponent

Example in binary:

• Original: 101.101
• Normalized: 1.01101 × 2^2

Since we always force the number to be 1.something, we know the leading digit is always 1!

Storage Optimization

Instead of wasting a bit storing the 1, we only store:

• The fraction part (the part after 1.)
• The exponent (in biased notation)

This gives us one extra bit of precision at the end of the fraction!

Representing Zero