Harmonic Mean Calculator

Definition

For positive real numbers a and b the two-argument harmonic mean is defined by H(a,b) = 2 / (1/a + 1/b). Equivalently H(a,b) = 2ab / (a + b). The first form stresses that H is the reciprocal of the arithmetic mean of the reciprocals; the second is convenient on paper. Both are identically equal when a ≠ 0 and b ≠ 0. On this page the domain is restricted to a > 0, b > 0 — the classical setting of the Pythagorean means and of rate averaging.

The general harmonic mean of n positive numbers x₁, …, xₙ is H = n / (Σᵢ₌₁ⁿ 1/xᵢ). For n = 2 the formula reduces exactly to the form used by this calculator. Mathematical model: the harmonic mean as one of the three Pythagorean means alongside the arithmetic and geometric means (Hardy–Littlewood–Pólya, Inequalities; Bullen, Handbook of Means and Their Inequalities; MathWorld — Harmonic Mean).

In teaching practice it helps to separate the mathematical identity from floating-point limits: this calculator shows the browser’s floating-point result, not a symbolic paper proof. When inputs are extreme, check the domain (a > 0, b > 0) first, then the floating-point range.

Historical and conceptual context

The Pythagorean means — arithmetic, geometric, and harmonic — were already known in ancient Greece in the theory of proportions and music (intervals, harmonic means of string lengths). In modern inequality theory the AM–GM–HM chain is a textbook staple (Hardy–Littlewood–Pólya). In statistics and finance the harmonic mean appears when averaging ratio-defined indicators (price–earnings, densities, throughputs) and when averaging speeds over equal distances.

Conceptually this calculator does not replace the geometric mean (multiplicative growth) or the arithmetic mean (additive quantities). It also does not compute a weighted harmonic mean Σ wᵢ / Σ (wᵢ/xᵢ). A longer list would use the n-term formula; this calculator deliberately has exactly two fields.

The speed motivation is the clearest classroom story: if you cover two equal legs at speeds a and b, total time is d/a + d/b, so average speed v̄ = 2d / (d/a + d/b) = H(a,b). The arithmetic mean (a+b)/2 would match equal time on both legs, not equal length — hence the classic trap “60 and 90 → 75 instead of 72”.

Formal properties

Domain: a > 0, b > 0. Range: H(a,b) > 0. Degree-1 homogeneity: H(λa, λb) = λ H(a,b) for λ > 0. Symmetry: H(a,b) = H(b,a). Monotonicity: if 0 < a ≤ a′ and 0 < b ≤ b′, then H(a,b) ≤ H(a′,b′).

AM–GM–HM inequality: for a, b > 0 one has (a+b)/2 ≥ √(a b) ≥ 2ab/(a+b), with equality ⇔ a = b. Thus HM is not always ≤ min(a,b) in a trivial sense that would exclude interior values (with 3 and 6, H = 4 lies between 3 and 6), but always H ≤ GM ≤ AM and min(a,b) ≤ H ≤ max(a,b).

Link to parallel resistors: the equivalent of two parallel resistors is R₁ ∥ R₂ = R₁R₂/(R₁+R₂) = H(R₁,R₂)/2. Series capacitors share the same reciprocal structure. That shows why the harmonic mean naturally describes quantities combined by summing reciprocals.

Full derivation

Step 1. From the arithmetic mean of reciprocals: A(1/a, 1/b) = (1/a + 1/b)/2.

Step 2. The harmonic mean is the reciprocal of that quantity: H = 1 / A(1/a, 1/b) = 2 / (1/a + 1/b).

Step 3. Algebra: 1/a + 1/b = (a+b)/(ab), so 2 / ((a+b)/(ab)) = 2ab/(a+b).

Step 4. Speed derivation: v̄ = total distance / total time = 2d / (d/a + d/b) = 2 / (1/a + 1/b) = H(a,b).

Step 5. HM ≤ GM follows from AM–GM on 1/a and 1/b: ((1/a)+(1/b))/2 ≥ √(1/(ab)); inverting (1/t decreasing on positives) yields H ≤ √(ab).

Special cases

Examples: a = 3, b = 6 → H = 4; a = 1, b = 3 → H = 1.5; a = 60, b = 90 → H = 72; a = 2, b = 2 → H = 2; a = 40, b = 60 → H = 48; a = 4, b = 9 → H = 72/13 ≈ 5.538462. When a = b one always has H = a = AM = GM.

An empty field leaves the placeholder (no result). Invalid text yields a parse error. a ≤ 0 or b ≤ 0 yields a domain error. The calculator no longer returns 0 when a + b = 0 — such a pair already falls outside a, b > 0.

Limits: as b → ∞ with fixed a > 0, H(a,b) → 2a; as b → 0⁺, H → 0⁺. That illustrates how a very small rate pulls the harmonic mean down hard.

Numerical notes

The calculator computes H = 2 / (1/a + 1/b), which is numerically equivalent to 2ab/(a+b) for moderate positive a, b in floating-point arithmetic. For extremely large a and b the product 2ab may overflow the floating-point range earlier than the sum of reciprocals; for tiny positives the reciprocals become large. For typical school and engineering values both paths are safe.

Comma and period are equivalent in the parser. Result formatting uses en-US / pl-PL locale and trims unnecessary zeros. The bar diagram scales AM, GM, and HM against the maximum of the three so the AM ≥ GM ≥ HM inequality is visible at a glance.

The calculator does not weight observations — both numbers enter with weight 1 in the n = 2 sense.

Related notions and sources

Related: geometric mean of two numbers (√(ab)), arithmetic mean of a list, weighted mean. In physics and electronics — parallel resistances and series capacitances. In finance — harmonic averaging of ratio indicators such as P/E for equal investment amounts.

Sources Hardy, Littlewood, Pólya — Inequalities (Cambridge); Bullen — Handbook of Means and Their Inequalities; Weisstein, Eric W. — “Harmonic Mean”, MathWorld; Wikipedia — Harmonic mean; NIST/SEMATECH e-Handbook (location measures in the context of means). CalcBoxer educational tool — the result does not replace a textbook.