Floating Point Representation in Computers

The Problem

Integers store whole numbers exactly. Real numbers like π, 0.1, or √2 often require infinite decimal or binary digits. Computers approximate using floating point — similar to scientific notation: a sign, a significand (mantissa), and an exponent.

Scientific Notation Analogy

123.45 = 1.2345 × 10². In binary float: significand × 2^exponent.

IEEE 754 Structure

Standard layout (single precision):

• 1 bit sign
• 8 bits exponent (stored with bias +127)
• 23 bits mantissa (fraction after implicit 1.)

Full details: IEEE 754 Explained

Precision Limits

Single precision offers ~7 decimal digits of precision. Double offers ~15. Large and tiny magnitudes have wider range but same relative precision limits.

Rounding and Representation Error

0.1 + 0.2 === 0.3  // false in JavaScript

0.1 has infinite repeating binary fraction — stored as nearest representable value.

Special Bit Patterns

• Zero: exponent and mantissa zero (±0 exists)
• Infinity: overflow result
• NaN: invalid operations (0/0)

Inspect exact bits with IEEE 754 Converter.

Best Practices

1. Avoid direct equality on floats — use epsilon tolerance
2. Use decimal types for currency (Java BigDecimal, Python Decimal)
3. Understand accumulation error in loops summing many floats
4. Know when to promote float to double

Fixed Point Alternative

Some embedded systems use fixed-point integers scaled by constant (e.g. store cents as integer). No exponent — predictable precision in known range.

Frequently Asked Questions

Why do games use float?

Hardware float ops are fast on GPUs/CPUs; sufficient for graphics coordinates.

What is machine epsilon?

Smallest difference between 1.0 and next representable float — ~1.19×10⁻⁷ for single.

Can I see float bits in Python?

import struct; struct.pack('f', 3.14) or use our converter.

Is double always better?

More precision and range but twice the memory; float enough for many apps.

How does this relate to integer binary?

Integers use two’s complement fixed point with implicit exponent 0 — exact within range.