What it does
A number is rational if it can be written as a fraction a/b of whole numbers. √2 cannot, and this page lets you see why in three ways:
- The fraction hunt. Pick a bottom number b and the page finds the top number a that makes a/b closest to √n. If a/b were exactly √n, then a² would equal n·b². Two bars compare them. For n = 2 they come within 1 of each other (7² = 49 and 2 × 5² = 50) but never meet, however big b gets.
- The digits. √n to 40 decimal places, worked out exactly. The decimal of every fraction either stops or repeats a block for ever; the digits of √2 never do.
- The proof. Step through the classic proof by contradiction: assume √p = a/b in lowest terms, square, and show that p divides both a and b, which cannot happen. A second proof shows that numbers such as 5 + 2√3 are irrational too.
It follows Class 10 Maths, real numbers (irrational numbers and their proofs) and Class 9, number systems (rational and irrational numbers, repeating decimals).
The proof leans on prime factors, which you can explore in HCF and LCM; √2 first turns up as the diagonal of a unit square in pythagoras proof playground.
How to use it
- In the first card set n (a whole number from 2 to 99) and slide b. Read a/b, a², n·b² and the gap between them.
- Try a perfect square such as n = 9: the bars meet at once, because √9 = 3.
- In the proof card pick √p is irrational and a prime p, then press Next step to reveal one step at a time, or Show all. Each step says what is claimed and, in grey, why it is true.
- Pick r + s√p is irrational, choose whole numbers r and s (s not 0) and step through again.
- Open Why does p divide a when p divides a²? to see the remainders of squares on dividing by p. None is 0.
Worked example
With n = 2 and b = 5, the closest top number is a = 7, because 5 × √2 = 7.07. Then a² = 49 and 2b² = 50: off by 1, so 7/5 = 1.4 is close to √2 = 1.41421356… but not equal. With b = 12, a = 17: 289 against 288, off by 1 again. With b = 29, a = 41: 1681 against 1682.
The proof for p = 2: suppose √2 = a/b with no common factor. Then 2b² = a², so 2 divides a², so 2 divides a. Write a = 2c: 2b² = 4c², so b² = 2c², and 2 divides b. Now 2 divides both a and b, which contradicts “no common factor”. So √2 is irrational.
Tips
- In a proof by contradiction, write the assumption clearly first (“suppose √p = a/b in lowest terms”), because the contradiction is with that sentence.
- The step “p divides a², so p divides a” needs p to be prime. It fails for 4: 4 divides 2² = 4, but 4 does not divide 2.
- For r + s√p, rearrange to get √p on its own; the other side is rational, which is the contradiction.
FAQ
Does a bigger b ever make the bars equal for n = 2? No. The proof shows no whole numbers a and b can make a² = 2b².
What about √6, which is not under a prime? √6 is irrational too. The same idea works with the prime 2 or 3 in its factors; the page’s hunt shows it never meets.
Is this from the textbook? No. The proof is the standard one from mathematics; the page and examples are written for fibub, which is not part of NCERT.
Is anything saved? No. Everything is worked out in your browser and nothing is stored or sent.
Where the numbers come from
- Standard results: the classical proof that the square root of a prime is irrational, closure of the rationals under arithmetic, and repeating decimals as fractions. Written for fibub.
- Digits of √n computed with exact whole-number square roots and checked against Python's decimal module to 40 places; best fractions checked against a brute-force search.
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Last updated 2026-11-09.