How mathematics proves that some numbers cannot be written as fractions

Introduction

Some numbers can be written as fractions—simple ratios of two integers—while others cannot. This article explains clearly and carefully why certain numbers are not expressible as fractions, how mathematicians prove that fact, and what it means for everyday numbers such as √2 and π. No specialist background is required.

What does “written as a fraction” mean?

When mathematicians say a number can be written as a fraction, they usually mean it is a rational number. A rational number is any number that can be expressed as p/q, where p and q are integers and q ≠ 0.

Examples of rational numbers: 1/2, -3/4, 7 (which is 7/1), and 0 (which is 0/1). Rational numbers include terminating decimals (0.25 = 1/4) and repeating decimals (0.333… = 1/3).

Numbers that cannot be written as such a ratio are called irrational numbers. Proving that a number is irrational means giving a logical demonstration that no pair of integers p and q can produce that number as p/q.

Why some numbers are irrational: the central idea

The central reason some numbers cannot be written as fractions is that their decimal expansions do not terminate or repeat in a pattern consistent with any ratio of integers. But rather than relying only on decimals, proofs use exact logical arguments to show a contradiction would arise if a number were rational.

These proofs often assume the opposite (that the number is rational), write it in lowest terms p/q, and then deduce an impossible consequence such as both p and q being even, contradicting lowest terms, or deriving other contradictions with integer properties. This style of reasoning is called proof by contradiction.

A classic example: why √2 is irrational

The simplest and most famous proof that some numbers are not fractions shows √2 is irrational. Here is a clear, step-by-step explanation.

Step 1. Suppose √2 is rational. Then there exist integers p and q (q ≠ 0) with √2 = p/q. We can assume p/q is in lowest terms—meaning p and q have no common factor greater than 1.

Step 2. Square both sides: 2 = p^2 / q^2, so p^2 = 2 q^2. This shows p^2 is even, so p must be even (an odd integer squared is odd).

Step 3. If p is even, write p = 2k for some integer k. Substitute back: (2k)^2 = 4k^2 = 2 q^2, so q^2 = 2 k^2. Thus q^2 is even, so q is even.

Step 4. Now both p and q are even, meaning they share a common factor 2. But this contradicts our assumption that p/q was in lowest terms. Therefore the assumption that √2 is rational is false. Hence √2 is irrational.

This proof is short, rigorous, and explanatory: it shows clearly why no integers p and q can satisfy the equation √2 = p/q.

Why the structure of the argument works

The √2 proof leverages the parity property: whether a number is even or odd. Squaring preserves parity in a way that lets us infer divisibility by 2. Because integers have well-understood divisibility properties, showing both numerator and denominator must be divisible by the same prime contradicts the lowest-terms assumption.

This strategy generalizes. For many numbers, you can assume a rational representation, transform the expression, and use number theory facts (parity, modular arithmetic, prime factorization) to reach a contradiction.

Other classic irrational numbers and proofs

√2 is only the beginning. Several other numbers are proven irrational by related methods.

√n for non-square integers n: If n is an integer that is not a perfect square (for example n = 3, 5, 6), then √n is irrational. The proof is similar: assume √n = p/q in lowest terms, square to get p^2 = n q^2, and inspect prime factors to conclude p and q share a prime divisor—contradicting lowest terms. This uses the unique factorization property of integers.

Cube roots and higher roots: If m is an integer that is not a k-th power, then the k-th root of m is irrational. The proof generalizes the parity argument by examining prime exponents in the factorization and showing exponents must be divisible by k, producing a contradiction with lowest terms.

π and e: Some important constants like π (the ratio of a circle’s circumference to its diameter) and e (the base of natural logarithms) are also irrational. Proving their irrationality is more advanced and uses analysis (infinite series, integrals, or continued fractions) rather than the elementary parity or prime-factorization methods. For π, a classic proof was given in the 18th and 19th centuries using properties of trigonometric integrals; for e, the proof uses its infinite series expansion and rational approximation arguments.

Decimal expansions: a helpful perspective

Decimal expansions provide an intuitive way to see the difference between rational and irrational numbers.

Rational numbers in base 10 always have decimal expansions that either terminate (like 0.5 = 1/2) or eventually repeat a finite block of digits (like 0.1666… = 1/6). This happens because there are only finitely many possible remainders when performing long division, so the process must cycle.

Irrational numbers have non-terminating, non-repeating decimal expansions. For example, √2 = 1.41421356… continues without settling into a repeating pattern. That non-repeating behavior is a reflection of the fact there is no finite algorithmic cycle of remainders that would produce a repeating decimal, equivalent to there being no integer ratio that yields the number.

Continued fractions: a deep characterization

Continued fractions give another useful way to tell rationals from irrationals. A simple continued fraction represents a number as a sequence of integer parts and reciprocals: a0 + 1/(a1 + 1/(a2 + 1/(a3 + …))).

Rational numbers have finite continued fraction expansions. Irrational numbers have infinite continued fraction expansions. Moreover, some irrationals, like quadratic irrationals (numbers that are solutions to quadratic equations with integer coefficients, for example √2), have periodic continued fraction expansions. This structure helps both to classify numbers and to compute good rational approximations.

Why proving irrationality matters

Establishing that certain numbers are irrational has both conceptual and practical importance.

Conceptually, it clarifies the landscape of real numbers by showing there are fundamentally different types of numbers: rationals and irrationals. It shows that rational numbers, dense though they are on the number line, form only a countable set while irrationals are uncountable and vastly more numerous.

Practically, irrationality influences numerical computation, approximation, and error analysis. For example, when designing algorithms that must represent constants like π, engineers and scientists rely on rational approximations but must be aware that these are only approximations, not exact values.

How mathematicians prove more complex numbers are irrational

For numbers beyond simple radicals, proofs use a wider set of tools from number theory and analysis:

Infinite series and rational approximations: For e, the infinite series 1 + 1/1! + 1/2! + 1/3! + … lets one show that if e were rational then a certain integer would lie strictly between two consecutive integers—impossible. The argument constructs ever-better rational approximations and derives contradictions.

Transcendence and special functions: Some numbers are not only irrational but transcendental, meaning they are not roots of any polynomial with integer coefficients (π and e are both transcendental). Proving transcendence uses more advanced techniques, such as the Lindemann–Weierstrass theorem for π. Transcendence implies irrationality but requires substantially deeper mathematics.

Diophantine approximation: This field studies how closely irrationals can be approximated by rationals and builds tools (like Roth’s theorem) that can show certain numbers cannot satisfy particular rational relationships.

Common misconceptions

Misconception: “All non-integer decimals are irrational.” This is false. Many decimals that look non-terminating are rational because they repeat; for example, 0.777… is 7/9.

Misconception: “If a number has a complicated decimal expansion, it’s probably transcendental.” Complexity of digits alone doesn’t prove transcendence. There are algebraic irrationals (solutions to polynomial equations) whose decimal expansions can be complicated yet are not transcendental.

Misconception: “There are more rational numbers than irrationals.” The opposite is true in a precise sense: rationals are countable (essentially listable), while irrationals are uncountable and therefore far more numerous on the number line.

Simple exercises to build understanding

Try these short exercises to practice the ideas:

1. Prove that √3 is irrational using the same style of proof as √2. Hint: assume √3 = p/q in lowest terms and square both sides.

2. Show that 0.121212… (repeating 12) is rational. Write it as a fraction by letting x = 0.121212…, then multiplying by 100 to shift the repeating block.

3. Explore continued fractions for √2: compute a few convergents (1, 3/2, 7/5, 17/12…) and see how they approximate √2 ever better. Notice the pattern repeats.

Broader context and historical notes

The discovery of irrational numbers dates back to ancient Greece. The Pythagoreans recognized that the diagonal of a unit square (√2) could not be expressed as a ratio of whole numbers, an insight that reportedly caused philosophical tension in their school because it challenged the belief that all quantities are ratios of integers.

Over centuries, mathematicians developed systematic ways to classify numbers, create rigorous proofs, and build the theory of real numbers. The formal distinction between rational, algebraic irrational, and transcendental numbers emerged as mathematics matured in the 18th and 19th centuries.

Practical takeaways

If you see a number like √5 or π, remember: these are not fractions of integers. That has practical consequences—when calculating with such numbers on a computer or with a calculator, you always use approximations (decimals or rational approximations), never exact finite fractions.

However, rational approximations can be extremely accurate. Engineers and scientists use fractional approximations or truncated decimal expansions to achieve whatever precision is needed for a task, even though the exact number may be irrational.

Conclusion

Some numbers cannot be written as fractions because assuming they can leads to logical contradictions with basic properties of integers. Simple, elegant proofs—like the classic demonstration that √2 is irrational—show why no integers p and q can produce these numbers as p/q. Beyond √2, mathematicians use prime factorization, infinite series, continued fractions, and advanced theorems to classify and prove the irrationality or transcendence of many important numbers, with both mathematical and practical significance.