No fraction p/q, with p and q integers, equals exactly √2. The proof is by contradiction: if such a fraction existed in lowest terms, both p and q would have to be even, and a fraction in lowest terms cannot have both even. What does exist are fractions very close to √2, such as 99/70, whose error is about 0.000072.
What it takes for √2 to be a fraction
A fraction is a quotient of two integers with a denominator that is not zero. A number that can be written this way is rational, as the guide on fractions and rational numbers explains. Asking whether √2 is a fraction is the same as asking whether such a pair of integers exists.
Before looking for one, you need to fix a form. Every fraction can be reduced by dividing the numerator and denominator by their greatest common divisor (GCD, also called the greatest common factor, GCF): 6/8 becomes 3/4 because GCD(6, 8) = 2. If you do not remember how to find it, the greatest common divisor tool calculates it. In lowest terms, p and q share no factor greater than 1. That is the condition the proof ends up breaking.
The proof by contradiction
You assume the opposite of what you want to prove and look for an absurd result.
- Assume the fraction exists. Let √2 = p/q, with p and q integers, q not zero, and GCD(p, q) = 1.
- Square both sides. This gives 2 = p²/q², that is, p² = 2q².
- Look at parity. The number p² is twice an integer, so it is even. An even square comes from an even base, so p = 2m for some integer m.
- Substitute. (2m)² = 2q² gives 4m² = 2q², and dividing by 2 leaves q² = 2m². So q² is even, and so is q.
With p and q both even, both are divisible by 2, which contradicts GCD(p, q) = 1. The assumption cannot hold.
The only divisibility fact used is that an even square has an even base. You can check it from the other side: if p = 2k + 1, then p² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd.
Lowest terms is not decoration. If fractions that were not reduced were allowed, “p and q are both even” would contradict nothing, because 2/4, 4/8 and 8/16 have the same value. The contradiction appears because the most reduced version was chosen from the start.
The argument also explains what happens with other roots. For √3 the same idea works with “multiple of 3” in place of “even”, but other roots, such as √8 or √12, need the argument adjusted. And with √4 no absurd result appears, because √4 = 2 = 2/1 is a fraction. The proof can only reach a contradiction when the radicand is not a perfect square.
Fractions that come close to √2
Since no fraction equals √2, the useful thing is to know which ones approximate it best for their size. The table gathers the first ones, with their squares and their absolute error compared with √2 ≈ 1.41421356 (rounded to 8 decimal places).
| Fraction | Exact square | Error (approx.) | Lands |
|---|---|---|---|
| 3/2 | 9/4 = 2.25 | 0.085786 | above |
| 7/5 | 49/25 = 1.96 | 0.014214 | below |
| 17/12 | 289/144 ≈ 2.0069 | 0.002453 | above |
| 41/29 | 1681/841 ≈ 1.9988 | 0.000420 | below |
| 99/70 | 9801/4900 ≈ 2.0002 | 0.000072 | above |
The error drops in each row and the side alternates, so every consecutive pair brackets √2. The errors and the decimal squares are rounded (to six and four decimal places); no square in the table is 2.
Worked example: how close 17/12 is
You want to know whether that fraction could equal √2 and, if not, by how much its square misses.
- Square the fraction. 17² = 289 and 12² = 144, so (17/12)² = 289/144.
- Compare with 2. Since 2 = 288/144, the difference is 289/144 − 288/144 = 1/144.
- Interpret. The square overshoots 2 by 1/144 ≈ 0.0069 (rounded to 4 decimal places), so the fraction is slightly larger than √2.
Result: 17/12 is not √2; its square is 289/144, which exceeds 2 by 1/144.
Check: the fraction ≈ 1.41667 and √2 ≈ 1.41421 (both rounded to 5 decimal places), so it lands about 0.0025 above, as the table shows.
There is a pattern that is not obvious at first glance. In all five fractions, p² − 2q² equals +1 or −1. For 7/5 it comes out as 49 − 50 = −1, and for the third one it is 289 − 288 = 1. That explains the side each one lands on and why the error cannot be large: the square of the fraction differs from 2 by exactly 1/q². If you continue the list, the next ones are 239/169 and 577/408. English Wikipedia, cited at the end, notes that an approximation equal to 577/408 appears in the ancient Indian Sulbasutras.
A common mix-up is to read 1.4142 as 14142/10000 and conclude that √2 is rational.
Where these fractions come from
The fractions in the table are the convergents of the continued fraction [1; 2, 2, 2, …], where the 2 repeats forever. Cutting the chain at a point and rebuilding the fraction gives each approximation: [1; 2] is 3/2, [1; 2, 2] is 7/5 and [1; 2, 2, 2] is 17/12. To see the process with a rational fraction, the continued fractions tool calculates the expansion and its convergents. It does not expand the infinite chain of an irrational number like √2, because it only accepts fractions.
If you start from a decimal and want the closest simple fraction, approximate decimal as fraction finds the best one with a maximum denominator that you choose and shows the error. With a maximum denominator of 70 and the decimal 1.41421356 it proposes 99/70, the fraction in the table; with a maximum of 100 it proposes 140/99, which is not in the table and has an error of the same order (0.000072). And the fraction roots calculator leaves √2 as an exact expression and marks the decimal with ≈. It gives an exact value only when the numerator and denominator are perfect squares, as in √(4/9) = 2/3.
An everyday case: A-series paper
A-series sheets, such as A4 and A3, are designed with sides in the ratio 1 : √2. If the short side is 1, the long side is √2. Folding the sheet in half across the long side, each half measures 1 by √2/2, and turned sideways its long-to-short ratio is 1 ÷ (√2/2) = 2/√2 = √2. The shape is preserved, which is why enlarging or reducing an A-series sheet distorts nothing. Here √2 appears as a ratio between lengths, never as a quotient of two integers.
Who discovered it
Tradition credits the discovery of irrationality to the Pythagorean school, and English Wikipedia says that little is known with certainty about when or how it happened, though the name of Hippasus of Metapontum is often mentioned. The attribution is uncertain: MacTutor records that the discovery is certainly attributed to the Pythagoreans but seems unlikely to be due to Pythagoras himself, and the Stanford Encyclopedia of Philosophy points out that Eudemus does not assign the Pythagorean discoveries reported by Proclus to Pythagoras in person. That is why this page speaks of the school and not of one person. The proof above uses parity, one of the shortest; classical presentations also give a geometric one.
Practice
Three checks
- Square 99/70 and say by how much it overshoots 2.
- Calculate p² − 2q² for 41/29.
- Explain why the proof does not work for √9.
Show the answers
- 99² = 9801 and 70² = 4900, so (99/70)² = 9801/4900. Since 2 = 9800/4900, it overshoots by 1/4900.
- 41² − 2 × 29² = 1681 − 1682 = −1.
- Because √9 = 3 = 3/1 is a fraction. In p² = 9q², p being a multiple of 3 leads to no contradiction.
Sources
- Wikipedia, Square root of 2: the name of Hippasus of Metapontum is often mentioned in connection with the discovery, and the Sulbasutras give an approximation equal to 577/408 (consulted October 17, 2026).
- MacTutor, Pythagoras of Samos: the discovery of irrationals is attributed to the Pythagoreans, but seems unlikely to be due to Pythagoras himself (consulted October 17, 2026).
- Stanford Encyclopedia of Philosophy, Pythagoras: Eudemus does not assign the Pythagorean discoveries reported by Proclus to Pythagoras himself (consulted October 17, 2026).