On this page
In Class 9 and 10 you learn that √2 is irrational and that the real numbers fill the number line with no gaps. But what exactly is √2, if it is not a fraction? For a long time even careful mathematicians leaned on the picture of a line. One of the first people to give a clean answer using only rational numbers was Richard Dedekind. Here is who he was, why the question bothered him, his construction worked through step by step, and what students can learn from his habits.
Who was Richard Dedekind?
The MacTutor biography of Dedekind, from the University of St Andrews, says he was born on 6 October 1831 in Braunschweig, Germany, and died there on 12 February 1916. He earned his doctorate at Göttingen in 1852 under Gauss and was Gauss's last pupil. He was the first to lecture on Galois theory at Göttingen, and he introduced the idea of an ideal in ring theory. In 1858 he moved to the Polytechnic in Zurich, where he began teaching in the autumn.
What problem was Dedekind trying to solve?
We can read his own account in the English translation by W. W. Beman, published in 1901 as Essays on the Theory of Numbers and now free on Project Gutenberg. In the preface to Continuity and Irrational Numbers, Dedekind says that in Zurich he had to lecture on the differential calculus for the first time and "felt more keenly than ever before the lack of a really scientific foundation for arithmetic".
The theorem that troubled him was that a quantity which keeps growing, but not beyond all limits, must approach a limiting value. To prove it, he says, "I had recourse to geometric evidences." He thought pictures were useful for teaching, but that "this form of introduction into the differential calculus can make no claim to being scientific". He resolved to keep thinking until he found a purely arithmetic foundation, and he writes: "I succeeded Nov. 24, 1858". He published the work in 1872.
What is a Dedekind cut?
Take all the rational numbers. Split them into two classes, A1 and A2, so that every number in A1 is less than every number in A2. Dedekind calls this a cut.
If the split happens at a rational number, such as 1/2, then that number is either the largest in A1 or the smallest in A2.
But some cuts have no rational number at the split. Dedekind's own example: let D be a positive whole number that is not a perfect square. Put in A2 every positive rational whose square is greater than D, and put everything else in A1. No rational number has square exactly D, so there is nothing at the boundary.
His move is to say that such a cut is itself a new number. "Whenever, then, we have to do with a cut (A1, A2) produced by no rational number, we create a new, an irrational number α, which we regard as completely defined by this cut (A1, A2)". With D = 2, the cut defines √2.
Worked example: the cut for √2 has no largest member
Let A1 hold all rationals that are negative, zero, or positive with square less than 2. To show the boundary is a real gap, we show that A1 has no largest element: for any positive p in A1, we can find a bigger q still in A1.
Use q = (2p + 2) / (p + 2).
q is bigger. q − p = (2p + 2 − p² − 2p) / (p + 2) = (2 − p²) / (p + 2). Since p² < 2, this is positive.
q is still in A1. q² − 2 = [(2p + 2)² − 2(p + 2)²] / (p + 2)² = (2p² − 4) / (p + 2)² = 2(p² − 2) / (p + 2)². Since p² < 2, this is negative, so q² < 2.
Try it. Start with p = 7/5 = 1.4, where p² = 1.96. Then q = (14/5 + 2) / (7/5 + 2) = (24/5) / (17/5) = 24/17 ≈ 1.41176, and (24/17)² = 576/289 ≈ 1.99308.
Again. From 24/17 you get 41/29 ≈ 1.41379, with square 1681/841 ≈ 1.99881. Then 140/99 ≈ 1.41414, with square 19600/9801 ≈ 1.99980.
The values climb towards √2 ≈ 1.41421 but never reach it, and none is the largest. That is the gap that the cut fills.
What mistakes do students make here?
Saying √2 = 1.414. That is an approximation. (1.414)² = 1.999396, not 2.
Thinking irrational means "infinitely long decimal". Some rationals, such as 1/3, also have endless decimals. Irrational means not a ratio of integers.
Mixing up the proof of irrationality with the definition. Your Class 10 textbook proves √2 cannot be a fraction. Dedekind's cut says what √2 is.
The NCERT Class 10 chapter on Real Numbers (reprint 2026-27) proves that √2 is irrational using the Fundamental Theorem of Arithmetic. Our post on real numbers for Class 10 walks through that proof.
If limits or irrational numbers still feel slippery, the Fermi Concept Library covers every concept in mathematics with its own lesson, so you can revise one idea at a time.
What did Dedekind believe about numbers?
In his 1888 essay The Nature and Meaning of Numbers, in the same Gutenberg volume, he opens with a line worth pinning above a desk: "In science nothing capable of proof ought to be accepted without proof." He also wrote that "numbers are free creations of the human mind". For the next step in the story of the real line, read about Georg Cantor and the sizes of infinity.
What can a student take from how Dedekind worked?
Notice when you are relying on a picture. He was comfortable using diagrams in teaching, but not in proofs.
Stay with a problem. He made a "fixed resolve" to keep thinking until he found an answer.
Teaching reveals gaps. It was explaining calculus to students that showed him what was missing.
Next step: use the formula q = (2p + 2)/(p + 2) starting from p = 1 and write down the first four fractions you get.
