What Mean Did You Mean?

Most of your math studies have focused on equations and identities… stuff on one side equal to stuff on the other side. Lots of big ideas in math, however, involve inequalities; where one thing is always less than another, or at least less than or equal. I was thinking about one flying back to the US (and boy, are my arms tired…. Ok, very bad, forgive please) and it offered the additional chance to introduce a notation you might not have seen yet.

Most of the time when you hear the word average or mean you think of them as the same thing. Add up all the numbers and divide by how many numbers there are. OK, the actual name of this is the Arithmetic Mean; but as far back as Pythagoras the Greeks had several others . There is a a Heronian mean (remember Heron and the formula for finding the area of a triangle given the three side lengths?)although the term sub-contrary mean was used (Pythagorus lived 500 years before Heron).Another ancient mean was the harmonic mean (this is the one you use to answer questions like what is your average speed if you drive to town at 60 mph and drive home at 40 mph..and the answer is NOT 50 mph), and even the standard deviation you studied in statistics units, is a type of mean, called the RMS, for Root-Mean-Square. But today, we want to talk about the arithmetic mean, and the geometric mean and a special inequality that ties them together.

If you remember arithmetic sequences, and geometric sequences, the big idea was that one was about addition (or subtraction) and the other about multiplication. That idea about the names extends to the arithmetic and geometric means. Ok, so let’s use a little formal notation, cause it feels so good. You know the arithmetic mean… \( \bar{x} = \frac{x_1 + x_2 + \cdots +x_n}{n}\) where x1 means the first number and x2 the second, etc…  \( \frac{\Sigma_{i=1}^{n} ({x_i})}{n}\)

The geometric mean works the same way, except instead of adding the numbers, we multiply them, and instead of dividing by n, we take the nth root. So for two numbers , say 2 and 8, the geometric mean would be (2*8)^1/2 = 4. If there were three numbers, we would take the cube root of their product, etc for more numbers. If that sounds like a screwy way to find the middle, let me point out one of the places it is useful. Suppose you have a triangle with sides of 3, 5, and 10. The perimeter is 18. What the arithmetic average gives you is the length of the sides if you wanted to make a triangle with the same perimeter and all sides equal, in this case we want each side to be 6.

If a rectangle has sides of 2 and 8 then its area is 16; but what if we wanted to make a rectangle with the sides all the same that had the same area. That would be a square, so we take the square root of 16 and get 4. The geometric mean of 2 and 8 is 4. We can do that with a solid too. Take a box with length, width and height of 2, 9, and 12 respectively. The volume is 216 cubic units. If we want a perfect cube with the same volume, we take the cube root of (2*9*12) and get 6. We say the geometric mean of 2, 9 and 12 is (216)^1/3. Of course we could do the same thing with any number of values, but the meaning in four-space or five-space etc, is a little harder to visualize. We even have a symbol for multiplying that allows us to write products in a condensed form similar to the sigma notation. The symbol we use is a capital Greek Pi. We can write  \( \Pi _{i=1}^{5} ({i+2})\)when we want to multiply (1+2) (2+2)(3+2)(4+2)(5+2). Notice we incremented through the numbers from 1 to 5 and added 2 to each, but instead of adding as we do with sigma notation, we multiply. Then we can write the geometric mean as  [(x₁)(x₂)…(x_n)]^1/n. 

If you remember how you can write a sequence in your TI-calculator and use the "sum" command to do the sigma notation, then you will be thrilled to know there is a command to find the product of the numbers in a string (look one line below the "sum" entry on your calculator.

So NOW we are ready to get to the big inequality we wandered through all that to get to; for any group of numbers, the geometric mean is always less than or equal to the arithmetic mean. Try a few simple ones, with two numbers, say 4 and 9, the geometric mean is 6, but the arithmetic mean is 6.5; or pick any three numbers of your choice and try it again.

For two numbers there is a really cool (is it still ok to say “cool”) geometric proof that is totally visual, and it gives me a chance to show you one more way that the geometric mean is MEANingful (couldn’t resist). If we take two lengths, call them a and b, and place them together on a single line, then M is the midpoint of a+b, and a+b/2 is the distance from m to either end of the line segment. But if we let the line comprising a and b be the diameter of a semi-circle, the perpendicular distance c from P, where a and b join to a point on the semicircle is the geometric mean of a and b, and r is the arithmetic mean. It is clear that r, being the hypotenuse of a right triangle, is always going to be greater than or equal to c, a leg. It is also easy to show that c = (a b)^1/2, the geometric mean of a and b. Since r=(a+b)/2, then the distance MP is r-b, which is (a-b)/2. So we have r² = c²+MP² . Substituting in we get (a+b)²/4 = c² + (a-b)²/4 so 4c²= (a+b)² –(a-b)² . The right side of this is a²+2ab+b² – (a²-2ab+b²) and when we subtract we are left with 4c² = 4ab, or just c² = ab, hence c=(ab)^1/2 and is the geometric mean of ab.

Later, I will try to show a really nice application of this inequality to solve the birthday problem.