The number whose discovery, legend says, got a man drowned.
The diagonal of a square with side 1 — can you write that length as a fraction?
The holes fractions can't fill on the number line — the irrational.
Experiment
Hands-on experiment
Where √2 was born — the diagonal of a square
Draw a square with side 1 and its diagonal has length √2. Watch where the fraction you are hunting sits on that diagonal.
The more you zoom, the tighter the fraction hugs √2. Yet the moment it lands exactly on the mark never comes.
Predict first — the diagonal of a unit square (√2 ≈ 1.414…). Can it be written exactly as a fraction?
Fraction hunt — find a fraction that squares to exactly 2
Each press tightens the net with a more precise fraction. Will (p/q)² ever hit exactly 2?
1/1
Current fraction
1.000000
Squared
1.000000
Gap from 2
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
The ancient Pythagoreans believed 'all is ratio (of whole numbers)'. Then their own theorem produced the square's diagonal, √2, and shattered that belief.
Assume √2 = p/q and a contradiction erupts (the even/odd argument). So rational numbers leave holes in the number line — holes so basic that even a diagonal's length falls through.
The numbers that fill those holes are the irrationals. Only rationals and irrationals together make a gapless number line (the reals) — the stage on which limits and calculus can stand.
Insight
Insights from the video
“√2 was mathematics' first scandal.”
The worldview 'everything is a ratio' collapsed from an inside discovery. Rather than bury the inconvenient truth, mathematics expanded its number system — why math is a science, not a religion.
“Irrationals aren't the exception — they're the majority.”
Drop a random point on the number line and it lands on an irrational with probability 1. Our familiar fractions are the vanishingly rare dust.
Misconception
Common misconceptions
The 1.41421356 on my calculator is √2.
That's only the opening of √2. Its decimal expansion never ends and never repeats — any finite string of digits is an approximation, not √2 itself.
If the decimals go on forever, the number is irrational.
0.333… goes on forever but equals 1/3 — rational. The test isn't infinity; it's repetition. Infinite AND non-repeating is what makes a number irrational.
Formula
Writing it as math
Write the hunt's failure in mathematical language and you get the definition of irrationality.
Definition of irrational
A number no ratio of integers can express. As a decimal it never terminates and never repeats.
Proof by contradiction, in one line
If p²=2q², then p is even, which forces q even too — destroying the 'lowest terms' assumption. This is the proof of why your hunt always failed.
Reals = rationals + irrationals
Only together do they fill the number line without gaps. That gaplessness (completeness) is the foundation of limits and calculus.
In Real Life
Where you meet it in real life
The secret of A4 paper
A-series paper has aspect ratio 1:√2 — the only ratio preserved when you fold the sheet in half. An irrational number hiding in your stationery drawer.
Camera apertures (f-stops)
f/1.4, f/2, f/2.8… each stop is a factor of √2, so the light doubles exactly per stop. The scale was engraved with an irrational number.
Equal temperament in music
One semitone is a frequency ratio of ¹²√2 — dividing the octave into 12 equal irrational steps is what made transposing to any key possible.
π and the world of circles
π is irrational too. Even the distance a wheel rolls in one turn can't be written exactly as a fraction.
Try Yourself
Test yourself
Q1Is 0.121212… (12 repeating) rational or irrational?Show answer ▾
Rational — it's 12/99 = 4/33. Any repeating infinite decimal converts to a fraction. The criterion for irrationality is not 'infinite' but 'never repeating'.
Q2Is √9 irrational?Show answer ▾
No — √9 = 3, a natural number. A radical sign doesn't make a number irrational; what matters is whether the value can be written as a fraction. √4 and √2.25 (=1.5) are rational too.
Q3What is √2 × √2? Is the product of two irrationals always irrational?Show answer ▾
√2 × √2 = 2 — rational! Irrationals can add or multiply into rationals (√2 + (−√2) = 0 as well). The irrationals are not closed under arithmetic.
Try answering yourself before revealing it — getting it wrong is where learning starts.
Connection
Concepts connect
Previous concept
Repeating Decimals
Every repeating decimal turned out to be a fraction — the next question opens the irrationals.
← Repeating Decimals labLeads to next
Prime Numbers
You've seen the gaps in the number line; now look at the atoms inside the naturals — the multiplicative bricks that build every number: primes.
Go to the Prime Numbers lab →Related
Labs worth exploring together
Related lab
Limits
The hunt's convergence is exactly a limit — √2 is defined as the limit of a sequence of rationals.
Go to the Limits lab →Related lab
Pythagorean Theorem
The culprit that produced √2 — see where the diagonal's length came from.
Go to the Pythagorean Theorem lab →