Irrational Numbers

Irrational Numbers

Real numbers that are not rational. Irrational numbers include numbers such as Square root of 6 , (2 − √29) / 4 , The square root of 5 , π, e, etc.

Nested diagram of number sets: natural, whole, integers, rationals, algebraics, reals, plus separate pure imaginary and...

Key Formula

x \text{ is irrational if } x \neq \frac{a}{b} \text{ for any integers } a, b \text{ with } b \neq 0

Where:

$x$ = A real number being tested for irrationality
$a$ = Any integer (the numerator)
$b$ = Any nonzero integer (the denominator)

Example

Problem: Prove that √2 is irrational.

Step 1: Assume, for contradiction, that √2 is rational. Then it can be written as a fraction a/b in lowest terms, where a and b are integers with no common factors and b ≠ 0.

\sqrt{2} = \frac{a}{b}, \quad \gcd(a,b) = 1

Step 2: Square both sides to eliminate the square root.

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

Step 3: Since a² = 2b², the number a² is even. An integer whose square is even must itself be even, so a = 2k for some integer k.

a = 2k \implies a^2 = 4k^2

Step 4: Substitute a² = 4k² back into the equation a² = 2b².

4k^2 = 2b^2 \implies b^2 = 2k^2

Step 5: Now b² is also even, so b is even. But both a and b being even contradicts our assumption that gcd(a, b) = 1. Therefore √2 cannot be written as a fraction, so it is irrational.

\text{Contradiction: } \gcd(a,b) \neq 1. \quad \therefore \sqrt{2} \text{ is irrational.}

Answer: √2 is irrational because assuming it equals a/b in lowest terms leads to a contradiction — both a and b would have to be even.

Another Example

This example focuses on classifying numbers rather than proving irrationality, giving practice with the key rule: a decimal that terminates or repeats is rational; one that does neither is irrational.

Problem: Determine whether each number is rational or irrational: (a) √9, (b) √5, (c) 0.3333…, (d) 0.101001000100001…

Step 1: Evaluate √9. Since 9 is a perfect square, √9 = 3, which is an integer and therefore rational.

\sqrt{9} = 3 = \frac{3}{1} \quad \text{→ Rational}

Step 2: Evaluate √5. Since 5 is not a perfect square, √5 = 2.2360679… The decimal never terminates or repeats. It is irrational.

\sqrt{5} = 2.2360679\ldots \quad \text{→ Irrational}

Step 3: Look at 0.3333… This is a repeating decimal. Any repeating decimal can be expressed as a fraction.

0.\overline{3} = \frac{1}{3} \quad \text{→ Rational}

Step 4: Look at 0.101001000100001… The number of zeros between each 1 keeps increasing, so the pattern never repeats in a fixed cycle. This makes it irrational.

0.101001000100001\ldots \quad \text{→ Irrational}

Answer: (a) Rational, (b) Irrational, (c) Rational, (d) Irrational.

Frequently Asked Questions

What is the difference between rational and irrational numbers?

A rational number can be expressed as a fraction a/b where a and b are integers and b ≠ 0. Its decimal either terminates (like 0.75) or eventually repeats (like 0.333…). An irrational number cannot be written as any such fraction, and its decimal expansion continues forever without a repeating block.

Is π (pi) an irrational number?

Yes. π = 3.14159265… has a decimal expansion that never terminates and never repeats. This was proven rigorously by Johann Lambert in 1761. Common approximations like 22/7 or 3.14 are close but not exact — they are rational numbers that approximate the irrational value of π.

Can you add or multiply two irrational numbers and get a rational number?

Yes. For example, √2 + (−√2) = 0, which is rational. Similarly, √3 × √3 = 3, a rational number. So the sum or product of two irrational numbers is not necessarily irrational — it depends on the specific numbers involved.

Irrational Numbers vs. Rational Numbers

	Irrational Numbers	Rational Numbers
Definition	Real numbers that cannot be expressed as a/b (integers a, b with b ≠ 0)	Real numbers that can be expressed as a/b (integers a, b with b ≠ 0)
Decimal form	Non-terminating, non-repeating	Terminating or eventually repeating
Examples	√2, π, e, √7	1/3, 0.75, −4, 0.1̄6̄
Countability	Uncountable — vastly more numerous	Countable — can be listed in a sequence
On the number line	Fill in the 'gaps' between rationals	Dense — between any two rationals lies another

Why It Matters

Irrational numbers appear constantly in geometry (π for circles, √2 for the diagonal of a unit square) and in algebra whenever you solve equations like x² = 5. Understanding them is essential for working with radicals, the Pythagorean theorem, and trigonometry. They also form the foundation for the real number line — without irrational numbers, the number line would have infinitely many "holes."

Common Mistakes

Mistake: Thinking that a long decimal like 0.142857142857… is irrational because it 'looks' like it goes on forever.

Correction: This decimal repeats the block 142857, so it equals 1/7 and is rational. The key test is whether the decimal eventually settles into a repeating cycle, not just whether it is long.

Mistake: Assuming √n is always irrational.

Correction: √n is only irrational when n is not a perfect square. For example, √16 = 4, which is rational. Always check whether the number under the radical is a perfect square before classifying.

Related Terms

Real Numbers — The set containing both rational and irrational numbers
Rational Numbers — The complementary subset of reals — numbers expressible as fractions
Surd — An irrational root such as √2 or ∛5
Transcendental Numbers — Irrational numbers that are not roots of any polynomial with integer coefficients
Algebraic Numbers — Numbers (rational or irrational) that are roots of polynomials with integer coefficients
Natural Numbers — The counting numbers — a subset of rational numbers
Integers — Whole numbers and negatives — all rational, none irrational
Imaginary Numbers — Not on the real number line; a separate concept from irrational numbers