Numerical approximation and error

Mathematical statement

For approximation a and reference x, absolute error is |a−x|. If x≠0, relative error is |a−x|/|x|. For tolerance τ≥0, being within tolerance means |a−x|≤τ.

Representative formula (exact TeX)

e=A(a,x)

Assumptions

• Exercises use exact rationals rather than treating displayed binary floating-point values as exact answers.
• Evaluate relative error only for a nonzero reference; truncation, rounding and data errors are not interchangeable.

Symbol scope

• e — absolute error
• a — approximation
• x — reference value

Intuition, definition, and misconceptions

Absolute error retains the original quantity’s scale, whereas relative error compares it with a nonzero reference magnitude. They answer different questions.

The bound |a−x|≤τ is equivalent to a∈[x−τ,x+τ], including both endpoints.

An error value alone does not identify its cause. Distinguishing discretization, truncation, rounding, measurement and model errors requires information about how the values were produced.

Do not evaluate relative error at a zero reference or call every error a rounding error.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real number — Approximations and reference values are real-valued.
• Absolute value — Error magnitude uses absolute value.
• Ratio and proportion — Relative error is a dimensionless ratio.

Verification boundary

Exact rational checks establish only the specified values and bounds; they assess neither algorithmic convergence nor a floating-point implementation.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add truncation/rounding sources, condition numbers, stability and convergence order.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.