Irrational Numbers

An irrational number is a number that cannot be expressed as the ratio of two integers.

Its decimal representation is infinite and non-repeating, meaning the digits after the decimal point continue endlessly without forming a regular pattern.

This lack of repetition makes it impossible to represent as a simple fraction of two integers, giving it a unique and somewhat mysterious quality.

For instance, the square root of 2 is an irrational number:

$$ \sqrt{2} = 1.414213562... $$

No matter how hard you try, you won’t find two integers \( a, b \in \mathbb{Z} \) that equal the square root of 2 when expressed as a fraction.

$$ \sqrt{2} = 1.414213562... \ne \frac{a}{b}, \ \ \ a,b \ \in \ \mathbb{Z} $$

This is because the decimal expansion of an irrational number is infinite and never follows a repeating pattern.

In short, irrational numbers cannot be represented as fractions of integers.

Why are they called irrational?

The term “irrational” refers to the fact that these numbers are not part of the rational number set, not because they are “crazy” or chaotic in nature.

The word "irrational" comes from the Latin irrationalis, meaning "not rational" or "not expressible as a ratio."

What is a rational number? A rational number is any number that can be expressed as the ratio of two integers, \( a/b \), with \( b \neq 0 \). Its decimal representation either terminates (e.g., \( 0.5 \) for \( 1/2 \)) or repeats indefinitely in a predictable sequence (e.g., \( 0.333... \) for \( 1/3 \)).

Famous Examples of Irrational Numbers

Some of the best-known irrational numbers include:

• The square root of 2 (\( \sqrt{2} \)): This number was first identified in ancient Greece when the Pythagoreans discovered that the diagonal of a square and its side could not have a rational ratio. The decimal expansion of \( \sqrt{2} \) is \( 1.414213562...\), and its digits continue indefinitely without repetition.
• Pi (\( \pi \)): Representing the ratio of a circle's circumference to its diameter, \( \pi \) is irrational (and transcendental, a more specialized category). Its value is approximately \( 3.141592653...\), with digits that appear random and never repeat.
• Euler's Number (\( e \)): Fundamental to calculus and exponential functions, \( e \) is another irrational number. Its decimal expansion starts as \( 2.718281828...\) and continues infinitely without periodicity.

These are just a few of the most famous examples.

In reality, irrational numbers are far more numerous—they are infinite.

Irrational Numbers in Everyday Life. While irrational numbers might seem confined to abstract mathematics, they have practical applications across many fields. For example, in geometry, \( \pi \) is essential for calculating areas and volumes of circular shapes. In physics, \( e \) is critical for modeling natural phenomena such as exponential growth and radioactive decay. In computer science, irrational numbers often emerge in high-precision calculations.

The Set of Irrational Numbers is Infinite

Irrational numbers are not rare exceptions among real numbers.

On the contrary, they are everywhere: between 0 and 1 alone, there are infinitely many irrational numbers. In fact, they are "more numerous" than rational numbers in a technical sense.

This concept, tied to set theory and cardinality, reveals the astonishing depth and richness of irrational numbers.

When combined with the set of rational numbers \( \mathbb{Q} \), the set of irrational numbers \( \mathbb{I} \) forms the larger set of real numbers \( \mathbb{R} \).

$$ \mathbb{I} \cup \mathbb{Q} = \mathbb{R} $$

As you can see, merging two infinite sets creates another infinite set, often of a "higher order."

The set of real numbers encompasses all other number sets: natural numbers \( \mathbb{N} \), integers \( \mathbb{Z} \), rational numbers \( \mathbb{Q} \), and irrational numbers \( \mathbb{I} \).

However, the real number set is not the largest... There is an even bigger set: the set of complex numbers \( \mathbb{C} \). But that's a topic for another time.

The Discovery of Irrational Numbers

The concept of irrational numbers was first recognized by the mathematicians and philosophers of the Pythagorean School, founded by Pythagoras of Samos in the 6th century BCE.

They discovered that the diagonal of a square with unit sides could not be expressed as the ratio of two integers.

For example, consider a square with a side length of \( l = 1 \).

Using the Pythagorean theorem, the length of the diagonal \( d \) can be calculated as follows:

$$ d^2 = 1^2 + 1^2 = 2 $$

Thus, the diagonal of the square is the square root of two:

$$ d = \sqrt{2} $$

The Pythagoreans realized this number could not be expressed as a ratio of two integers, making it irrational:

$$ \sqrt{2} = \frac{a}{b} $$

Where \( b \ne 1 \), since \( \sqrt{2} \notin \mathbb{N} \), meaning it is not a natural number.

If such integers existed, their squared ratio would equal 2:

$$ \frac{a^2}{b^2} = 2 $$

However, this is impossible, as it would imply \( a \) and \( b \) are not coprime, contradicting the initial assumption.

Thus, the square root of two was the first irrational number discovered by humanity.

Many others followed, including \( \sqrt{3}, \sqrt{5}, \sqrt{7} \), and in general, any \( \sqrt{n} \) where \( n \) is not a perfect square, along with numbers like \( \pi \), and more.

The discovery of irrational numbers deeply unsettled the Pythagoreans, who believed the universe was governed by numerical proportions and that everything could be represented as a ratio of integers. When one member, likely Hippasus of Metapontum, proved the square root of 2 could not be expressed as a fraction, it was considered heretical. Some accounts suggest Hippasus was punished, exiled, or even killed for revealing the existence of irrational numbers.

In conclusion, irrational numbers, with their infinite and non-repeating digits, open the door to a mathematical universe that defies intuition and pushes the boundaries of rational thought.

As the German mathematician Leopold Kronecker once said: "God created the natural numbers; everything else is the work of man."

Yet irrational numbers seem to belong to an even higher realm, where chaos and order intertwine in an endless dance.