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PROOF 186

Prove.

Significant figures do not increase under multiplication

On the real numbers ℝ.

A way to read this proof

First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.

Definition D1

A measured value is written x̃ = x(1 + δ). Here x is the true value and δ the relative error. What the measurement guarantees is |δ| ≤ ε, and this ε ≥ 0 is called the relative error bound.

Proof.

The true value x and the actual error δ are unknowable. What is known is the single bound ε.convention

∎

Definition D2

Having n significant figures is the digit-count expression of the relative error bound ε ≤ 5 × 10⁻ⁿ.

Proof.

(1)Rounding to n digits from the first significant figure, the error does not exceed half the last digit — at most 5 × 10⁻ⁿ in relative error.convention
(2)Each additional significant figure divides ε by 10. The digit count is a scale marking the size of ε.(1)

∎

Lemma L1

The relative error of the product x̃ỹ is δ₁ + δ₂ + δ₁δ₂.

Proof.

(1)Write x̃ = x(1 + δ₁), ỹ = y(1 + δ₂).D1
(2)x̃ỹ = xy(1 + δ₁)(1 + δ₂)(1), A3, E
(3)(1 + δ₁)(1 + δ₂) = 1 + δ₁ + δ₂ + δ₁δ₂A4, A2
(4)x̃ỹ = xy(1 + (δ₁ + δ₂ + δ₁δ₂))(2), (3), E
(5)This is the form of D1, so the relative error of the product is δ₁ + δ₂ + δ₁δ₂.(4), D1

∎

Theorem T1

The relative error bound of the product is ε₁ + ε₂ + ε₁ε₂.

Proof.

(1)|δ₁ + δ₂ + δ₁δ₂| ≤ |δ₁| + |δ₂| + |δ₁δ₂|L1, A6
(2)|δ₁δ₂| = |δ₁||δ₂|A7
(3)|δ₁| ≤ ε₁, |δ₂| ≤ ε₂D1
(4)|δ₁||δ₂| ≤ ε₁ε₂(2), (3), A7, A9
(5)|δ₁ + δ₂ + δ₁δ₂| ≤ ε₁ + ε₂ + ε₁ε₂(1), (3), (4), A8, A11
(6)Therefore ε₁ + ε₂ + ε₁ε₂ is a relative error bound for the product.(5), D1

∎

Theorem T2

This bound cannot be reduced. No guaranteed relative error bound falls below ε₁ + ε₂ + ε₁ε₂, and in particular none is smaller than max(ε₁, ε₂).

Proof.

(1)Only |δᵢ| ≤ εᵢ is known, so the case δ₁ = ε₁, δ₂ = ε₂ cannot be excluded.A10
(2)In that case the relative error of the product is ε₁ + ε₂ + ε₁ε₂ — the bound of T1 is actually attained.(1), L1, T1
(3)0 ≤ ε₂ and 0 ≤ ε₁ε₂, so ε₁ = ε₁ + 0 + 0 ≤ ε₁ + ε₂ + ε₁ε₂.D1, A9, A5, A1, A8
(4)The same computation holds for ε₂. The bound is not smaller than max(ε₁, ε₂).(2), (3)

∎

Corollary C

The significant figures of a product do not exceed those of the less precise factor.

Proof.

(1)The digit count of significant figures is fixed by the relative error bound. The factor with the larger ε is the one with fewer digits.D2
(2)The relative error bound of the product is not smaller than max(ε₁, ε₂).T2
(3)Therefore the digit count of the product does not exceed that of the factor with the larger ε — the one with fewer significant figures.(1), (2), D2

∎

Remark

Can A10 be removed from this proof?

The bound itself (T1) and its attainability (T2) are different claims. Find where A10 is actually used.