The square root of 2 is irrational

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

The diagonal (die Diagonale) of a square with side 1 is \( \sqrt{2} \). This number cannot be written as a fraction: it is irrational. Plus a look at history – from Babylon to the Pythagoreans.

📄 Open worksheet (PDF, German)

The video

Watch the colours: the diagonal and \( \sqrt{2} \) are orange, the small squares are blue.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

The most important points

Four squares with side 1 make a big square with area 4. The four diagonals form a tilted square made of four half squares – area 2. Its side is the diagonal. So the diagonal is \( \sqrt{2} \approx 1.41 \).

The tilted square has area 2. Its side is the diagonal \sqrt{2} . — The tilted square has area 2. Its side is the diagonal \( \sqrt{2} \).

\( 1.4^2 = 1.96 \), \( 1.41^2 = 1.9881 \), \( 1.414^2 = 1.999396 \) – almost 2, but never exactly. Fractions never hit it exactly either: \( \left(\frac{7}{5}\right)^2 = 1.96 \), \( \left(\frac{17}{12}\right)^2 \approx 2.007 \).

The proof: if \sqrt{2} = p/q were fully reduced, p and q would both be even – a contradiction (der Widerspruch). — The proof: if \( \sqrt{2} = \frac{p}{q} \) were fully reduced, \( p \) and \( q \) would both be even – a contradiction (der Widerspruch).

Rational and irrational

Rational: can be written as a fraction, e.g. \( \frac{1}{2} = 0.5 \) or \( \frac{1}{3} = 0.333\ldots \) (the 3 repeats). Irrational: no fraction possible, the digits never end and never repeat, e.g. \( \sqrt{2} = 1.41421356\ldots \). If a natural number is not a square number, its square root is irrational.

Babylon (about 1700 BC): clay tablet with \sqrt{2} \approx 1.414212 . Greece (about 500 BC): the Pythagoreans and Hippasus. — Babylon (about 1700 BC): clay tablet with \( \sqrt{2} \approx 1.414212 \). Greece (about 500 BC): the Pythagoreans and Hippasus.

The Pythagoreans believed: “Everything is number.” The Pythagorean Hippasus is said to have discovered that side and diagonal do not fit into any ratio of whole numbers. A legend says he gave away the secret and drowned in the sea. Historians today consider the legend unlikely.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

Show the worksheet solutions

Try it yourself

Task

Rational or irrational? a) \( \sqrt{25} \) b) \( \sqrt{7} \) c) \( \frac{2}{3} \) d) \( \sqrt{50} \)

Show solution

a) rational (5) b) irrational c) rational d) irrational

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: the square root of 2 is irrational (diagonal in the square, rational and irrational numbers). Ask me five short questions, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Write in simple German and put an English translation in brackets after each sentence.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.