Geometric Mean Calculator

Definition

The geometric mean of two non-negative numbers a and b, written G(a, b) or √(a·b), is the square root of their product. Equivalently, for a > 0 and b > 0, G = exp((ln a + ln b)/2) — the arithmetic mean of the logarithms mapped back by the exponential. The calculator computes √(a·b) and, for positive factors, shows the logarithmic check.

The definition is standard in inequality analysis (Hardy, Littlewood, Pólya, Inequalities, Cambridge University Press) and on MathWorld (Geometric Mean). The extension to n positive numbers is G = (x₁·…·xₙ)^(1/n). This page handles only n = 2.

A geometric reading: √(a·b) is the side of a square whose area equals that of a rectangle with sides a and b (geometric-mean theorem / similarity). In finance the same form appears for an average growth factor: if capital is multiplied by r₁ and r₂, the mean factor is √(r₁ r₂).

Historical and conceptual context

The geometric mean was known in Greek geometry (mean proportional of segments). Cauchy’s Cours d'analyse (1821) relates means; the modern AM–GM statement with equality at equal summands was cemented in Hardy–Littlewood–Pólya (1934/1952). Bullen (Handbook of Means and Their Inequalities, Springer, 2003) organizes the power-mean family, where the geometric mean is the limit case of exponent 0.

Conceptually G measures a multiplicative center of scale, not an additive shift. That is why it fits growth rates, indices, and ratios, and not raw temperatures on a scale with an arbitrary zero. The arithmetic mean stays appropriate for sums and shifts; the harmonic mean fits speeds in the equal-leg model 2ab/(a+b).

This calculator does not replace multi-period CAGR or a Fisher index; it shows only G of two non-negative numbers and the AM comparison on the same pair. For classroom checks, recomputing √(a·b) by hand on the worked examples (4 and 9 → 6; 2 and 8 → 4) confirms that the live panel matches the definition exactly.

Formal properties

Calculator domain: a ≥ 0, b ≥ 0. Codomain: [0, ∞). Commutativity: G(a, b) = G(b, a). Homogeneity: G(λa, λb) = λ G(a, b) for λ ≥ 0. If a = b, then G = a.

AM–GM inequality: for a, b > 0 one has (a+b)/2 ≥ √(a·b), with equality ⇔ a = b. Together with the harmonic mean: H ≤ G ≤ AM for a, b > 0, where H = 2ab/(a+b).

Logarithm link: ln G = (ln a + ln b)/2 when a, b > 0. So G is the arithmetic mean on the log scale, mapped back by exp.

Monotonicity: for fixed b > 0 the map a ↦ √(a b) increases on [0, ∞).

Full derivation

Step 1 (definition). Take G² = a·b with G ≥ 0, hence G = √(a·b) on a, b ≥ 0.

Step 2 (AM–GM). Let a, b > 0. Then (√a − √b)² ≥ 0 ⇒ a − 2√(ab) + b ≥ 0 ⇒ (a+b)/2 ≥ √(ab).

Step 3 (equality). (√a − √b)² = 0 ⇔ √a = √b ⇔ a = b (non-negative roots).

Step 4 (exp/ln form). For a, b > 0: ln(a·b) = ln a + ln b, so √(a·b) = exp(½ ln(a·b)) = exp((ln a + ln b)/2).

Step 5 (computation). √(a·b) in floating-point arithmetic; if a = 0 or b = 0 the result is 0. Negative a or b are rejected before the square root so the user is not confused by NaN or by a positive √ from a both-negative pair.

Special cases

Examples: G(4, 9) = 6; G(2, 8) = 4; G(1, 1) = 1; G(0, 9) = 0; G(4.5, 4.5) = 4.5; G(1, 4) = 2; G(16, 25) = 20. The pair (−4, 9) → domain error.

When a = b, the AM and G bars have the same height. When one factor tends to 0 with the other fixed, G tends to 0 faster than AM.

This page does not compute ⁿ√(x₁…xₙ) for n ≠ 2 and does not compute a weighted geometric mean with exponent weights.

Numerical notes

The product a·b can overflow the floating-point range for extreme inputs; then the result is out of range. For a, b > 0 the form exp((ln a + ln b)/2) can be gentler for very large or very small factors — the panel compares both forms when they are finite.

Comma and period are equivalent. The calculator does not return a complex mean or the square root of a negative product.

Educational result: matches √ in the browser, not a CAS that simplifies √(4·9) symbolically to 6 before floating point — here 6 comes out exactly.

Related notions and sources

Related: arithmetic mean, harmonic mean, AM–GM inequality, CAGR / mean growth factor, power means (Bullen). Neighboring calculators: arithmetic mean of a list, harmonic mean of two numbers, weighted mean.

Sources: Hardy, Littlewood, Pólya — Inequalities (CUP); Bullen — Handbook of Means and Their Inequalities (Springer, 2003); MathWorld — Geometric Mean; Wikipedia — Geometric mean. Educational tool.