Editing Harmonic mean (section)

==={{anchor|Harmonic mean of two numbers}}Two numbers===
[[File:MathematicalMeans.svg|thumb|right|A geometric construction of the three [[Pythagorean means]] of two numbers, ''a'' and ''b''. The harmonic mean is denoted by ''H'' in purple, while the [[arithmetic mean]] is ''A'' in red and the [[geometric mean]] is ''G'' in blue. ''Q'' denotes a fourth mean, the [[quadratic mean]]. Since a [[hypotenuse]] is always longer than a leg of a [[right triangle]], the diagram shows that <math>H \le G \le A \le Q</math>.]]
[[File:harmonic_mean_graphical_computation.svg|thumb|A graphical interpretation of the harmonic mean, ''z'' of two numbers, ''x'' and ''y'', and a [[nomogram]] to calculate it. The blue line shows that the harmonic mean of 6 and 2 is 3. The magenta line shows that the harmonic mean of 6 and −2 is −6. The red line shows that the harmonic mean of a number and its negative is undefined as the line does not intersect the ''z'' axis.]]
For the special case of just two numbers, <math>x_1</math> and <math>x_2</math>, the harmonic mean can be written as:<ref name="wolfram"/>
:<math>H = \frac{2x_1 x_2}{x_1 + x_2} \qquad </math>    or    <math> \qquad \frac{1}{H} = \frac{(1/x_1) + (1/x_2)}{2}.</math>
(Note that the harmonic mean is undefined if <math>x_1 + x_2 = 0</math>, i.e. <math>x_1 = -x_2</math>.)

In this special case, the harmonic mean is related to the [[arithmetic mean]] <math>A = \frac{x_1 + x_2}{2}</math> and the [[geometric mean]] <math>G = \sqrt{x_1 x_2},</math> by<ref name="wolfram"/>
:<math>H = \frac{G^2}{A} = G\left(\frac{G}{A}\right).</math>

Since <math>\tfrac{G}{A} \le 1</math> by the [[inequality of arithmetic and geometric means]], this shows for the ''n'' = 2 case that ''H'' &le; ''G'' (a property that in fact holds for all ''n''). It also follows that <math>G = \sqrt{AH}</math>, meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means.