Irrational Numbers — Mathematics Knowledge System

Overview

Irrational numbers are real numbers that cannot be expressed as a ratio p/q of two integers with q nonzero. Their decimal expansions neither terminate nor repeat. Famous examples include sqrt(2), pi, and e. Together with the rationals they fill out the entire real line, yet in a precise sense 'most' real numbers are irrational.

Intuition

Imagine trying to measure the diagonal of a 1-by-1 square with a ruler that only has rational markings. No matter how fine you make the markings, you will never land exactly on the diagonal's length of sqrt(2). Equivalently, the decimal expansion of sqrt(2) = 1.41421356... goes on forever without any repeating block. Rational numbers are like music with a repeating pattern; irrational numbers are like an infinite melody that never loops.

Formal Definition

Definition

A real number x is irrational if it is not rational, i.e., there do not exist integers p and q with q > 0 such that x = p/q. Equivalently, x is irrational if and only if its decimal expansion is non-terminating and non-repeating. The set of irrationals is the complement of Q in R: it equals R \ Q.

x \in \mathbb{R} \setminus \mathbb{Q} \iff \nexists\, p,q \in \mathbb{Z},\; q \neq 0,\; x = \frac{p}{q}

Definition of irrational number

\sqrt{2} = 1.41421356\ldots \quad (\text{non-terminating, non-repeating})

Decimal expansion of sqrt(2)

Notation

Notation	Meaning
$\mathbb{R} \setminus \mathbb{Q}$	The set of all irrational numbers
$\sqrt{2}$	Square root of 2, the classic irrational number
$\pi$	Ratio of circumference to diameter; transcendental irrational
$e$	Euler's number; base of the natural logarithm; transcendental irrational

Theorems

Theorem 1: Irrationality of sqrt(2)

sqrt(2) is irrational. Proof by contradiction: assume sqrt(2) = p/q in lowest terms (gcd(p,q)=1). Then 2q^2 = p^2, so p^2 is even, hence p is even; write p = 2k. Then 2q^2 = 4k^2, so q^2 = 2k^2, making q even. But then gcd(p,q) >= 2, contradicting the assumption.

Theorem 2: Density of Irrationals

Between any two distinct real numbers there exists an irrational number. Equivalently, the irrationals are dense in R.

Theorem 3: Uncountability of Irrationals

The set of irrational numbers R \ Q is uncountable. Since Q is countable and R is uncountable, their difference must be uncountable. In the measure-theoretic sense, almost every real number is irrational (the rationals have Lebesgue measure zero).

Theorem 4: Algebraic vs. Transcendental

An irrational number is algebraic if it is a root of a nonzero polynomial with rational coefficients (e.g., sqrt(2) satisfies x^2 - 2 = 0). It is transcendental otherwise. Both pi and e are transcendental. The algebraic irrationals are countable, so transcendental numbers form the 'vast majority' of reals.

Worked Examples

1
Assume for contradiction that sqrt(3) = p/q where p, q are integers with gcd(p,q) = 1 and q > 0.
2
Squaring both sides gives 3q^2 = p^2.
$3q^2 = p^2$
3
This means p^2 is divisible by 3, so p must be divisible by 3 (since 3 is prime). Write p = 3k.
4
Substituting: 3q^2 = 9k^2, so q^2 = 3k^2, meaning q is also divisible by 3.
$q^2 = 3k^2$
5
But then gcd(p,q) >= 3, contradicting our assumption. Therefore sqrt(3) is irrational.

✓ Answer

sqrt(3) is irrational, proved by contradiction via the same parity argument as for sqrt(2).

Practice Problems

Easyproof writing

Prove that sqrt(5) is irrational.

EasyMultiple choice

Which of the following is a transcendental number?

Easyfree response

Is the product of two irrational numbers always irrational? Give a proof or counterexample.

Mediumproof writing

Prove that log_2(3) is irrational.

Common Mistakes

Common Mistake

sqrt(4) is irrational because it involves a square root.

sqrt(4) = 2, which is a rational integer. Not every square root is irrational -- only square roots of non-perfect-squares are irrational.

Common Mistake

0.101001000100001... is rational because it looks like it has a pattern.

Having a visual pattern is not enough; the decimal must eventually repeat the same block. This number has a pattern but no repeating period, so it is irrational.

Common Mistake

The sum of two irrational numbers is always irrational.

sqrt(2) + (-sqrt(2)) = 0, which is rational. Sums of irrationals can be rational.

Quiz

Historical Background

The discovery of irrational numbers is attributed to the Pythagoreans around 500 BCE, when Hippasus of Metapontum showed that the diagonal of a unit square cannot be expressed as a ratio of whole numbers. This was philosophically shocking to a school that believed 'all is number' (meaning rational number). Hippasus is said to have been drowned for revealing the secret. The rigorous theory of irrational numbers had to wait until the 19th century, when Dedekind (Dedekind cuts, 1872) and Cantor gave precise constructions of the reals.

500 BCE
Hippasus discovers irrationality of sqrt(2)
Hippasus of Metapontum
300 BCE
Euclid provides a proof that sqrt(2) is irrational in the Elements
Euclid
1761
Lambert proves pi is irrational
Johann Heinrich Lambert
1844
Liouville constructs the first explicit transcendental numbers
Joseph Liouville
1873
Hermite proves e is transcendental
Charles Hermite
1882
Lindemann proves pi is transcendental
Ferdinand von Lindemann
1891
Cantor's diagonal argument shows the reals are uncountable, so 'most' reals are irrational
Georg Cantor

Summary

An irrational number is a real number that cannot be written as p/q for integers p, q with q nonzero.
The classic proof that sqrt(2) is irrational uses contradiction and divisibility by 2.
Irrationals include algebraic numbers like sqrt(2) and transcendental numbers like e and pi.
The set of irrationals is uncountable and has full Lebesgue measure in R -- 'most' real numbers are irrational.

References

BookNiven, I. (1956). Irrational Numbers. Mathematical Association of America.
BookHardy, G. H. & Wright, E. M. (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press.
WebsiteWeisstein, E. W. Irrational Number. MathWorld.