Why Do We Use the Real Numbers?

Let’s quickly recall a few basic sets of numbers:

$\mathbb{N}= \{ 0, 1, 2, 3,...\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textbf{natural numbers}.$
$\ \mathbb{Z}= \{ ...-3,-2,-1 ,0, 1, 2, 3,...\} \ \ \ \ \ \ \ \ \ \ \ \ \textbf{integers}.$
$\mathbb{Q}= \{ \frac{a}{b}\ |\ a,b\in \mathbb{Z}\textbf{ and } b\ne 0\} \ \ \textbf{rational numbers}.$

In particular, the set of rational numbers consists of all fractions, where the numerator (number on top) and denominator (number on bottom) are integers (and the denominator is not 0). Here are some examples of rational numbers:

$\frac{1}{2},\ \ \ \ \frac{2}{3},\ \ \ \ \frac{-17}{26},\ \ \ \ \frac{5}{1},\ \ \ \ \frac{0}{3}$

If the denominator is 1, we will identify the rational number with the integer consisting of the numerator alone. If the numerator is 0, we will identify the rational number with the integer 0. For example, we have:

$\frac{5}{1}=5,\ \ \ \ \frac{0}{3}=0$

At first glance, it would appear that the rational numbers would be sufficient to solve all “real world” problems. So, why do we use the real number system? The real number system is so complicated to understand that teachers in elementary school, middle school, and even in high school do not even attempt to define it rigorously.

Well, it turns out that a long time ago, a group of people called the Pythagoreans showed that the rational numbers alone are actually not enough to solve all real world problems. The issue was first discovered when applying the now well-known Pythagorean Theorem.

The Pythagorean Theorem: In a right triangle with legs of lengths a and b, and a hypotenuse of length c, c²= a²+ b².

Proving the Pythagorean Theorem is not too difficult and in fact, I have given an elementary proof here: Proving the Pythagorean Theorem

So, how does the Pythagorean Theorem tell us that the rational numbers are not enough? The answer lies in the following simple picture:

Question: In a right triangle where both legs have length 1, what is the length of the hypotenuse?

Let’s try to answer this question. If we let c be the length of the hypotenuse of the triangle, then by the Pythagorean Theorem, we have c² = 1² + 1² = 1 + 1 = 2. Since c² = c ⋅ c, we need to find a number with the property that when you multiply that number by itself you get 2. The Pythagoreans showed that if we use only rational numbers, then no such number exists.

Theorem: There does not exist a rational number a such that a² = 2.

The proof of this theorem, although not too difficult, does require a bit of mathematical knowledge. Over the next few weeks, I will be teaching you the mathematics necessary to prove this theorem, with the ultimate goal of supplying a proof.

As it turns out, there is a real number a such that a² = 2 (in fact, there are two such real numbers). In order to prove this, we would first need to give a rigorous definition of the real numbers. This requires a lot more work than one would initially expect.

The details of this definition are developed in my math books Pure Mathematics for Beginners, Set Theory for Beginners, and Topology for Beginners. These books contain no prerequisites and are perfect for anyone just starting out in theoretical math. You can get them all for one low price by clicking on the image below.

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