What is an Irrational Number? Definition, Examples & Properties

Irrational Number - Definition

An irrational number is a real number that cannot be written as the ratio of two integers. In formal terms, a real number x is irrational if there are no integers p and q (with q ≠ 0) such that x = p/q. The decimal expansion of an irrational number is non-terminating and non-repeating.

Examples include √2 = 1.41421356237…, π = 3.14159265358…, and e = 2.71828182845… None of these decimals end. None of them settle into a repeating pattern.

The Symbol for Irrational Numbers

Irrational numbers do not have a single universally adopted symbol. Three notations are commonly used:

The notation R \ Q (or ℝ \ ℚ) is preferred in formal mathematics because it states exactly what the set is — every real number that is not rational. The symbol P is more common in school textbooks and is easier to typeset.

How to Identify an Irrational Number

A number is irrational if it satisfies any one of the following four checks:

1. It cannot be written as p/q. If no integer fraction equals the number, it is irrational.

2. Its decimal expansion does not terminate and does not repeat. A decimal like 0.333… is repeating (rational). A decimal like 0.10110011100… is non-repeating (irrational).

3. It is the square root of a non-perfect square. √2, √3, √5, √7, and √11 are all irrational.

4. It is a known irrational constant. π, e, and the golden ratio φ are always irrational.

The table below applies these checks to common numbers.

Examples of Irrational Numbers

The most commonly cited irrational numbers are √2, π, e, and the golden ratio. Square roots of non-perfect squares form a broader category that includes infinitely many irrationals.

√2 (Square Root of Two)

Decimal expansion: 1.41421356237… The diagonal of a unit square has length √2. This was the first number proven to be irrational, by the Pythagoreans in the 5th century BC.

π (Pi)

Decimal expansion: 3.14159265358… π is the ratio of any circle's circumference to its diameter. The fraction 22/7 is a common approximation but is not equal to π — 22/7 = 3.142857… (a repeating rational).

e (Euler's Number)

Decimal expansion: 2.71828182845… e is the base of the natural logarithm. It appears in compound interest, exponential growth and decay, and probability theory. It is named after the Swiss mathematician Leonhard Euler.

φ (Golden Ratio)

Decimal expansion: 1.61803398874… φ equals (1 + √5) / 2. It appears in geometry, art, architecture, and certain growth patterns in nature.

Square Roots of Non-Perfect Squares

If a positive integer n is not a perfect square, then √n is irrational. So √2, √3, √5, √6, √7, √8, √10, √11 — and infinitely many others — are all irrational.

Proof That √2 is Irrational

The irrationality of √2 can be shown by a proof by contradiction. The argument assumes the opposite of what is to be proven, derives a contradiction, and concludes that the original assumption must be false.

Step 1. Assume √2 is rational. Then √2 = p/q where p and q are integers, q ≠ 0, and p/q is in lowest terms (p and q share no common factor other than 1).

Step 2. Square both sides: 2 = p² / q².

Step 3. Rearrange: p² = 2q².

Step 4. Since p² equals 2 times an integer, p² is even. If p² is even, then p must also be even.

Step 5. Let p = 2k for some integer k. Substitute into Step 3: (2k)² = 2q², which gives 4k² = 2q², or q² = 2k².

Step 6. Since q² equals 2 times an integer, q² is even. So q must also be even.

Step 7. Both p and q are even, meaning they share a common factor of 2. This contradicts Step 1, where p/q was in lowest terms.

Step 8. The contradiction means the original assumption is false. Therefore √2 cannot be written as p/q. √2 is irrational.

The same reasoning extends to √3, √5, √6, and the square roots of all non-perfect squares.

Properties of Irrational Numbers

Irrational numbers obey their own arithmetic rules, distinct from those of rationals.

• Sum of a rational and an irrational is always irrational. Example: 2 + √3 is irrational.

• Product of a non-zero rational and an irrational is always irrational. Example: 5 × √2 is irrational.

• Sum of two irrationals can be rational or irrational. Example: (2 + √3) + (2 − √3) = 4, which is rational. But √2 + √3 is irrational.

• Product of two irrationals can be rational or irrational. Example: √2 × √2 = 2, which is rational. But √2 × √3 = √6, which is irrational.

• The set of irrationals is not closed under addition or multiplication. Closure means an operation on members of a set always produces another member of the same set. The set of irrationals fails this — adding or multiplying two irrationals can produce a rational.

• The decimal expansion of an irrational number is always non-terminating and non-repeating. This is the defining numerical property of the set.

• The reciprocal of a non-zero irrational is irrational. If x is irrational and x ≠ 0, then 1/x is also irrational.

Rational vs Irrational Numbers

Rational and irrational numbers together form the set of real numbers. They differ on every important feature.

Every rational number is a real number. Every irrational number is also a real number. No number is both rational and irrational.

Common Confusions About Irrational Numbers

Several misconceptions appear repeatedly when students first encounter irrational numbers. Each is addressed below.

1. "All fractions are rational." False. A fraction is rational only if both its numerator and denominator are integers (and the denominator is not zero). The expression √2/3 is a fraction, but its numerator is irrational — so the fraction itself is irrational.

2. "All non-terminating decimals are irrational." False. A non-terminating decimal is irrational only if it also fails to repeat. The number 0.333… is non-terminating, but it repeats — it equals 1/3, which is rational. Only non-terminating and non-repeating decimals are irrational.

3. "22/7 equals π." False. The fraction 22/7 = 3.142857142857… is a repeating rational. π = 3.14159265358979… is non-repeating and irrational. 22/7 is an approximation that agrees with π to two decimal places — no further.

4. "All square roots are irrational." False. The square root of a perfect square is rational. √4 = 2, √9 = 3, √16 = 4, and √100 = 10 are all rational. Only square roots of non-perfect squares are irrational.

Surds and Irrational Numbers

A surd is an irrational number expressed using a root symbol — square root, cube root, or higher. Every surd is irrational, but not every irrational number is a surd.

π and e are irrational but cannot be expressed as the root of any rational number. Such numbers are called transcendental — a stricter category that includes π and e but excludes √2 (which is irrational but algebraic).

Where Irrational Numbers Appear in the Curriculum

Irrational numbers are formally introduced once students have mastered rational numbers, decimals, and square roots.

In each curriculum, irrational numbers appear after rational numbers and lead into work with real numbers, surds, and (later) the real number line.

A Brief History of Irrational Numbers

Did You Know? The irrationality of √2 was first proven by the Pythagorean Hippasus of Metapontum in the 5th century BC, while studying the diagonal of a unit square. The result contradicted the Pythagorean belief that every quantity could be expressed as a ratio of integers.

Indian mathematicians, including Manava (c. 750 BC) and Brahmagupta (628 AD), worked with surds long before they were formally accepted in European mathematics. Irrational numbers were not rigorously defined as real numbers until the 19th century, through the work of Richard Dedekind and Georg Cantor.