﻿ Proofs from The Book - The most elegant axioms, theorems, and proofs in school mathematics.

## Proof that the Product of Two Rational Numbers is Rational

The purpose of this blog is to introduce mathematical proofs as early as possible. When I say mathematical proof, it does not have to be rigorous; sometimes simple reasoning by words alone would suffice especially in the early grades.

In this post, I am going to show you a simple proof that can be probably given as a problem for middle school students. That is, when the students already understand the definition (and meaning) of rational numbers.

A rational number is a number which can be expressed as a fraction whose numerator and denominator are both integers and the denominator is not equal to 0. A student will probably argue that the fractions of integral numerators and denominators, but it is not the case. The fraction $\frac{\pi}{2}$ (not the image above) for instance is a fraction, but not a rational number. From that definition, we can show that the product of two rational numbers is rational. I have written the proof as layman as possible. Continue Reading →

11. December 2013 by Guillermo Bautista

## A Proof of the Arithmetic Mean Geometric Mean Inequality

Have you ever wondered what is the relationship between the arithmetic mean and the geometric mean of two numbers? Or you have probably heard some theorems about their relationship. In this post, we are going to see that their relationship is really very easy to imagine if represented geometrically.

The arithmetic mean of two numbers $a$ and $b$ is $\frac{a+b}{2}$, while their geometric mean is $\sqrt{ab}$. Now, we represent them using the figure below.

7 December 2013, Created with GeoGebra

Consider the semi-circle with diameter $\overline{AB}$ and radius $\overline{EF}$. We construct $\overline{CD} \perp \overline{AB}$ anywhere such that $C$ is on the circle and $D$ is on the diameter.If we let the length of $\overline{AB} = a$ and $\overline{BD} = b$, then the radius $\overline{EF} = (a+b)/2$.

## Proof That The Sum The First N Odd Integers is a Square

The sum of the first n odd integers is a square. This is a theorem and can easily be proven if you have already learned proof by mathematical induction. Even though it sounds like a boring theorem, it is actually very interesting if represented visually. Consider the following diagrams and their numerical representations.

$1 + 3 = 2^2$

$1 + 3 + 5 = 3^2$

Can you see the pattern? Let’s have some more examples.  Continue Reading →

05. December 2013 by Guillermo Bautista