Geometric Mean of Two Numbers

The geometric mean is the number that sits between a and b multiplicatively: a/G = G/b, so G² = ab. Worked example: the geometric mean of 4 and 25 is √100 = 10, and the ratios 4:10 and 10:25 are both 2:5. Use it whenever the quantities being averaged multiply rather than add — an investment up 100% one year and down 50% the next has geometric mean √(2 × 0.5) = 1, correctly reporting no net gain, while the arithmetic mean of +100% and −50% misleadingly says +25%.

Euclid's Elements VI.13 constructs it with ruler and compass: lay a and b end to end, draw a semicircle on the total as diameter, and the perpendicular from the join has length √(ab) — a proof that the geometric mean of two positive numbers never exceeds their arithmetic mean, since the perpendicular can never be longer than the radius. That is the AM–GM inequality in a single picture. The same construction is the "altitude on the hypotenuse" relation of right triangles, and it fixes the golden ratio and the geometric-mean level in a musical octave. Trap: the two numbers must share a sign. The geometric mean of −4 and −25 is 10 by the algebra, but of 4 and −25 it does not exist in the reals.