# Floating-point number representation and arithmetic

We've been using a representation of the real numbers without knowing or thinking much about it.

A sensible scheme for computing in the real numbers (as opposed to integers)?

Rationals are dense in the reals: how about mayfly model in exact arithmetic?

Approximate representation in 64 bits

how big?

how distributed?

## IEEE 754 floating point standard

sign, exponent, mantissa

machine epsilon, min, max

toy system,
toy system with denormalization

### Floating point arithmetic

catastrophic loss of significance

Exercise: plot ((1+x)-1)/x on
remedies