A picture I shall never forget! The proof of the Arithmetic-Geometric Mean Inequality.

A simple google search of the term ‘Arithmetic Geometric Mean Inequality Proof‘ will give you numerous proofs. Probably, school mathematics would have taught one of them. If you were extremely lucky, you might have got the chance and freedom to play around with the inequality and think of a few original ideas to prove it.

For those who are wondering what this is all about, let me just state the inequality and make it pretty clear what we are trying to prove.

The ‘Arithmetic Geometric Mean Inequality’ states that, given any 2 numbers a and b, the arithmetic mean is always greater than or equal to the geometric mean.

Just stop for a while and try out a few numbers. Ask some questions. Is this always true? Why is it so? What are the different ways to prove this? Is there a counter-example?

When the Hungarian mathematician George Polya came across this problem, he casually drew a picture as the proof of this, which many mathematicians say they would never forget.

How could this simple picture of a circle with three line segments possibly explain the AGM inequality?

Let’s break this down.

This is a circle of diameter (a+b). A line perpendicular to the diameter is drawn from the point where lengths a and b are separated.

A simple calculation (using similarity of triangles) will show that the length x is √ab, the square root of product ab. 

Now, we also know that the radius of the circle is (a+b)/2. The picture below makes it very clear that the radius is always greater than the length x. Moreover, it can also be clearly seen that the equality holds only when a and b are equal.

This is really beautiful. It not proves the result by some manipulation of terms, but makes it very clear why this is true, and its truly aesthetic and elegant.

Can you think of more ways to prove this?

What our Math classrooms need is more ideas, opportunities to play, to experience the beauty of Math and to think original ideas.