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Cantors Schlange: ℕ und ℚ sind gleich groß

About this app

Cantors Schlange frisst sich durch das unendliche Bruchgitter, hängt jedem gekürzten Bruch ein Nummernschild aus ℕ an und überspringt Doppelte wie 2/4; unten landen die Brüche auf einem immer dichteren Zahlenstrahl. Gegenmodus „zeilenweise“ läuft für immer in der ersten Zeile davon, „Ganz ℚ“ nummeriert im Reißverschluss 0, +, −. Bruch antippen: Die Schlange rast hin und zeigt seine Nummer.

With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.

Why ℚ is countable

All positive fractions fit into an infinite grid: column p and row q hold p/q. Cantor’s enumeration does not go row by row but zigzags through the diagonals on which p + q is constant. Every diagonal is finite, so every fraction is reached after finitely many steps. Fractions that are not reduced, such as 2/4 = 1/2, are skipped. So every natural number gets exactly one fraction and every fraction exactly one number: a bijection from ℕ onto the positive rational numbers. With 0 and the negative fractions taking turns, this works for all of ℚ.

What the app shows

At the top lies the fraction grid that Cantor’s snake eats its way through. Every new fraction gets a number plate from ℕ, duplicates turn grey and point to their reduced twin. Below, the list number ↦ fraction grows, and every fraction flies as a dot onto the number line, which gets denser and denser between 0 and 1. The counter-mode counts row by row and runs off along the first row forever, so 1/2 is never reached. A third mode numbers all of ℚ like a zip: 0, plus, minus.

Try it yourself

Tap any fraction: the snake races there and a magnifier shows its number, for 4/6 the twin 2/3. Let the snake run fast and watch the number line get dense while every number is handed out exactly once. Prediction cards pause at exciting moments, such as before the first duplicate or when counting row by row, so a class can vote first. Under Practice there are questions to shout out, revealed in the scene.

Frequently asked questions

Why are the rational numbers countable?
Because they can be numbered without gaps. Cantor writes all fractions p/q in a grid and zigzags through the finite diagonals. Every fraction is reached after finitely many steps, duplicates such as 2/4 are skipped. This gives a bijection between ℕ and ℚ.
Aren’t there more fractions than natural numbers because ℚ is dense?
No. There is always another fraction between two fractions, but the size of a set only depends on whether a one-to-one correspondence exists. Cantor’s enumeration provides one, so ℕ and ℚ have the same cardinality. Only the real numbers have more than countably many elements.
Why can’t you count fractions row by row?
The first row 1/1, 2/1, 3/1 and so on is infinitely long. If you try to count it completely first, you never get to the second row, and fractions like 1/2 would never get a number. The diagonals, in contrast, are all finite.
What can I try in the app?
Let the snake run step by step or fast, tap fractions and see their number, spot duplicates, fail with counting row by row, number all of ℚ including 0 and negative fractions, and predict and reveal questions for a class.

Subject: Mathe ohne Zahlen

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