The Rational-Irrational Divide
Look past basic fractions and explore the boundary where clean ratios break down. Master decimal structures, algebraic proofs of irrationality, rational approximations, and the continuum of the real number line.
Like Duolingo, but for The Rational-Irrational Divide. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
21 levels across 3 sections, about 42 minutes end to end, roughly 8 days at five minutes a day. It moves through Pinning Down the Boundary; Rules of Interaction and Proof; and Approximations, Density, and the Continuum. It assumes you already know the basics.
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Key ideas in The Rational-Irrational Divide
- In base ten, terminating decimals require denominators whose only prime factors are 2 and 5.
- Any prime factor other than 2 or 5 in a reduced denominator forces an infinite repeating cycle.
- Multiplying by 10 raised to the cycle length shifts one complete repeating period across the decimal point.
- Subtracting the original decimal cancels out the infinite repeating tail completely.
- Solving the resulting equation yields an exact integer fraction ready for reduction.
- Long division by denominator n can only produce non-zero remainders from 1 up to n - 1.
- A remainder repeating triggers an identical digit cycle, bounding the period length to at most n - 1.
- A decimal is rational if and only if it eventually repeats a fixed, finite block of digits.
- Predictable non-repeating patterns still represent irrational numbers that no integer division can create.
- Rationality depends strictly on whether a quantity is a ratio of integers, not on its representation.
- Changing base alters whether digits terminate or repeat, but can never turn a rational number irrational.
- A radical yields a rational result when the radicand factors into an exact power of the root index.
- Non-perfect powers leave an irreducible radical component that makes the entire value irrational.
- Roots like the cube root of 64 yield 4, whereas the square root of 32 leaves irrational root 2 factors.
- Multiplying numerator and denominator by the radical moves irrationality to the top without changing value.
- Clearing the radical from the denominator simplifies 6 / sqrt(3) into the clean rational multiple 2 * sqrt(3).
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives The Rational-Irrational Divide the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
What happens if a fully reduced fraction's denominator contains a prime factor like 3 or 7?
Get it right to open this lesson and 20 more in the app.
Where The Rational-Irrational Divide takes you
- 1
Pinning Down the Boundary
- Reading Decimal Signatures
- Unmasking Disguised Numbers
- 2
Rules of Interaction and Proof
- Arithmetic Across the Boundary
- Proving Irrationality with Contradiction
- 3
Approximations, Density, and the Continuum
- Rational Approximations and Continued Fractions
- Density and the Structure of the Number Line
- Algebraic vs. Transcendental Numbers
3 sections · 7 units · 21 levels. Built to play, not to enroll.
You pick the voice
The Rational-Irrational Divide is taught in the The Professor style: clear, structured, thorough. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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