Existing in Mathlib · Number Theory
Irrationality of the Square Root of Two
The Real square root of 2 is not equal to any rational number.
- irrational numbers
- real square root
- prime numbers
- natural squares
- rational numbers
Exact theorem
Exact Mathlib statement
theorem irrational_sqrt_two : Irrational (√2)The theorem at a glance
Irrationality of the square root of two at a glance
A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.
Irrationality of the square root of two at a glance
Detailed visual description
The poster foregrounds Irrational (√2) and its plain-language meaning. A precise unit square contributes the diagonal motif; exact natural-square arrays compare one cell, a two-cell domino, and a four-cell two-by-two square; a medallion with exactly two counted marks records the prime specialization. Gold implication threads converge on an unlabeled emerald seal, while the checked-route copy and scope footer keep the exact specialized Real boundary explicit.
Statement structure
From hypotheses to conclusion
This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.
The square diagonal amid dense rational markers
Detailed visual description
The upper-left sequence contains one square cell, a nonsquare two-cell domino, and a two-by-two four-cell square, all built from congruent cells. Below them, a precise square carries a corner-to-corner gold diagonal and faint compass arcs. The diagonal continues to a large emerald diamond with a gold double frame on a dense horizontal field of many small cobalt circles and smaller diamonds. Cobalt marks occur immediately on both sides, so the composition introduces no rational-free interval, negative root, arbitrary-radicand claim, or quantitative approximation claim.
Why it matters
A mathematical landmark
The irrationality of √2 is a foundational result connecting elementary geometry, number theory, and the structure of the real numbers. Mathlib's compact declaration is also an instructive library landmark: a one-line specialization rests on a reusable characterization of irrational natural square roots and the fact that a prime natural is not a square.
ProofAtlas record
What has been checked
Mathlib is the source of the theorem; the local Lean replay and page review are separate.
Claim boundary
No new theorem is claimed
This page indexes Mathlib's exact specialized Real statement Irrational (√2): the principal Real square root of the natural numeral 2 is not equal to any rational number cast into the reals. The selected declaration does not itself state the general square-root criterion for every natural, integer, or rational radicand; it does not cover both signed square roots; and it does not assert transcendence or a quantitative rational-approximation bound.
- The selected declaration is specialized to the Real principal square root of 2; it is not itself a theorem about every nonsquare natural, integer, or rational radicand.
- The declaration does not assert that √2 is transcendental or provide a quantitative rational-approximation bound.
- The declaration uses Real.sqrt, so it does not state a claim about both signed solutions of x² = 2 or about complex square roots.
- ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.
Source and local evidence
Where the theorem comes from
- Existing declaration
irrational_sqrt_twoin mathlib- Relationship
- The checked artifact is the upstream declaration itself
- Local Lean evidence
artifact.library.mathlib.irrationality-sqrt-two.v001