Why Some Numbers Cannot Be Written as Fractions

Draw a square with sides exactly 1 unit long. Its diagonal has a perfectly definite length: √2, about 1.41421356. You can construct it with a ruler and compass. And yet no fraction, no matter how large its numerator and denominator, is exactly equal to it.

Numbers like this are called irrational. The discovery that they exist reportedly shocked the ancient Greeks, whose philosophy held that everything could be expressed as a ratio of whole numbers.

Irrational Numbers Aren't Rare, They're Almost Everything

It's natural to think of irrational numbers as a few exotic exceptions: √2, π, e. The truth is the reverse.

If you picked a number at random from the number line, the probability it would be a fraction is zero. Fractions are countable. They can be listed one by one. The real numbers can't be. In a precise mathematical sense, almost every real number is irrational. Fractions are the rare exceptions.

What "Rational" Means

A rational number is any number that can be written as a ratio of two integers:

p / q,   where q ≠ 0

Examples: 3/4, −7/2, 5 (which is 5/1), and 0.333... (which is 1/3).

An irrational number is a real number that can't be written this way.

The Decimal Test

Every rational number has a decimal that either terminates or eventually repeats:

1/4  = 0.25
1/7  = 0.142857142857…   (repeats "142857")
5/12 = 0.41666…          (repeats "6")

Why? When you divide p by q with long division, each remainder must be between 0 and q − 1. After at most q steps, a remainder has to repeat, and from then on the digits cycle. See repeating patterns on the division tables.

Irrational numbers have decimals that never end and never repeat:

√2 = 1.41421356237309504880…
π  = 3.14159265358979323846…

The Proof That √2 Is Irrational

This proof is over 2,000 years old and still one of the most elegant in mathematics. It's a proof by contradiction.

1. Suppose √2 = p/q, a fraction in lowest terms (p and q share no common factor).
2. Square both sides: 2 = p²/q², so p² = 2q².
3. So p² is even. The square of an odd number is odd, so p must be even. Write p = 2k.
4. Substitute: (2k)² = 2q², so 4k² = 2q², and q² = 2k².
5. So q² is even, meaning q is even too.
6. But now p and q are both even, sharing a factor of 2. That contradicts step 1.

So √2 can't be written as a fraction. The same argument shows √n is irrational whenever n isn't a perfect square. Compare the square numbers list with the square roots list: only perfect squares have whole-number roots.

An Insider Reference: The Legend of Hippasus

According to ancient legend, the discovery of irrational numbers came from Hippasus of Metapontum, a member of the Pythagorean school in the 5th century BC. The Pythagoreans believed "all is number," meaning whole numbers and their ratios.

The story goes that Hippasus revealed that some lengths couldn't be expressed as ratios, and was drowned at sea for the betrayal. Historians treat the drowning as myth; the sources are late and contradictory. But the mathematical shock was real. The Greeks responded by building geometry on lengths and proportions rather than numbers, an approach set out in Euclid's Elements.

Other Famous Irrationals

π and e are also transcendental: they aren't the root of any polynomial equation with integer coefficients. Ferdinand von Lindemann proved this for π in 1882, which finally showed that "squaring the circle" with a ruler and compass is impossible. Explore the digits with the first 1,000 digits of pi.

Rational Approximations

We can't write irrational numbers as fractions, but we can get very close:

The approximation 355/113 was found by Chinese mathematician Zu Chongzhi in the 5th century AD. It's accurate to 6 decimal places, and no fraction with a smaller denominator comes closer. These best approximations come from continued fractions.

Two Concepts Worth Knowing

Proof by Contradiction

Assume the opposite of what you want to prove, then show it leads to an impossibility. It's the standard way to prove a number is irrational, since "can't be written as a fraction" is hard to prove directly.

Countable vs. Uncountable

The rational numbers can be listed in a sequence, so they're countable. The real numbers can't, so they're uncountable. That's the precise reason "almost all" real numbers are irrational.

Quick Answer: Why Can't Some Numbers Be Written as Fractions?

Some numbers are irrational: no ratio of two integers equals them exactly. For √2, assuming it equals a fraction p/q in lowest terms forces both p and q to be even, a contradiction. Rational numbers always have terminating or repeating decimals, while irrational numbers like √2 and π never end or repeat.

Try Them Yourself

• Square Roots List: spot which roots are irrational
• Square Numbers List: the perfect squares with whole-number roots
• First 1,000 Digits of Pi: look for a repeating pattern (there isn't one)
• Division Tables: repeating decimals from fractions
• Number Theory Formulas: divisibility and parity rules
• The Meaning of Pi: the most famous irrational number

Divide 1 by 7, then by 13, then by 17 using long division. Count how many digits each takes to start repeating. You'll see why every fraction must eventually loop.