Multiplication of Fractions Proof

Let me show you a pretty cool proof. We will prove that:

1/2 • 3/4 • 5/6 • 7/8 • 9/10 • ... • 99/100 < 1/10

With most proofs, it is easy to see if the conjecture is correct on face, but the actual proof takes more work. In this example, it is extremely difficult to tell if 1/10 would be less or greater than the product.

Let's start by denoting the left-hand side with A.

A = 1/2 • 3/4 • 5/6 • 7/8 • 9/10 • ... • 99/100

We will also denote the sort of opposite of A with B.

B = 2/3 • 4/5 • 6/7 • 8/9 • 10/11 • ... • 98/99

You will notice that every term in the B product has a corresponding term in the A product. 1/2 and 2/3 are both the first term, 3/4 and 4/5 are both the second term, and so on. You will also notice that every term in the B fraction is greater than every term in the A fraction. So, we can conclude that B is greater than A, or A is less than B.

A < B

Let's multiply both sides of that inequality by A.

A • A < A • B

A² < AB

The product of A and B is something that we can figure out pretty easily. Since the numerators of A are the denominators of B and vice versa, everything will cancel out. All that will be left is 1/100, since 1 and 100 were not in the B product. So, AB = 1/100. Substitute 1/100 in for AB to get:

A² < AB
A² < 1/100

If we solve this inequality for A, we will find that A < 1/10.

A² < 1/100
√(A²) < √(1/100)
A < 1/10

But what is A equal to? At the beginning, we denoted it to be equal to the left-hand side of the hypothesis inequality. Substitute that in for A, and we get:

1/2 • 3/4 • 5/6 • 7/8 • 9/10 • ... • 99/100 < 1/10

And there is our proof. These types of proofs don't really innovate mathematics, but I do think they can be fun.