MAT 300

 

Several Pythagorean means

 

Arithmetic mean

The arithmetic mean of two numbers exceeds the smaller number by the same as the larger number exceeds the mean.   I.e., A is the arithmetic mean of a and b, a<b, if  

A  a = b  A.

For example, 15 is the arithmetic mean of 12 and 18 since  15  12 = 18  15.  The modern approach is to compute the arithmetic mean by using the formula 

Geometric mean

The geometric mean of two numbers is the number such that the difference of the geometric mean and the smaller number divided by the difference of the larger number and the geometric mean is equal to the smaller number divided by the geometric mean.  I.e.,  G is the geometric mean of a and b, a<b, if

Equivalently, in our modern definition, G is the geometric mean of a and b if the ratio of the first number to the geometric mean is the same as the ratio of the mean to the second, or a:G = G:b.  I.e.,

 

So 6 is the geometric mean of 2 and 18 since

Harmonic mean

The harmonic mean of two numbers is the number such that the difference of the harmonic mean and the smaller number divided by the difference of the larger number and the harmonic mean is equal to the smaller number divided by the larger number.  I.e., if H is the harmonic mean of a and b, a<b, then

 So, 8 is the harmonic mean of 6 and 12, since

Alternatively,  we can compute the harmonic mean H of two numbers a and be using the following formula:

or