The Hidden Structure of Fractions
Explore the surprising deeper architecture of rational numbers far beyond elementary arithmetic. Discover how mediants, Farey sequences, continued fractions, and binary trees organize the rationals into a rich geometric system.
Like Duolingo, but for The Hidden Structure of Fractions. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
A short one: 12 levels across 2 sections, about 24 minutes end to end, roughly 5 days at five minutes a day. It moves through Navigating the Rational Landscape and Approximation and Decomposition.
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Key ideas in The Hidden Structure of Fractions
- The mediant of two positive fractions a/b and c/d is (a+c)/(b+d)
- If a/b < c/d, then a/b < (a+c)/(b+d) < c/d holds strictly
- The mediant does not equal the arithmetic mean unless both denominators are identical
- Two fractions a/b < c/d are adjacent in a Farey sequence if and only if their cross-product difference bc - ad equals 1
- For 5/8 and 7/11, bc - ad = (8 * 7) - (5 * 11) = 56 - 55 = 1, but their mediant 12/19 or another fraction with a smaller denominator like 2/3 would intervene if both exist, wait: 5/8 = 0.625, 7/11 = 0.6363... 8*7 - 5*11 = 1. Actually bc - ad = 1 means they ARE unimodular neighbors and appear consecutively in F_11!
- Let's test a non-neighbor pair instead to test the false claim: 4/7 and 5/9 gives 7*5 - 4*9 = 35 - 36 = -1 (so 5/9 < 4/7, with 4*9 - 5*7 = 1). Let's take 3/8 and 5/12: 8*5 - 3*12 = 40 - 36 = 4 != 1. True or false: The fractions 3/8 and 5/12 can appear consecutively as neighbors in a Farey sequence of some order n. (False)
- The coordinates (0,0), (b,a), and (d,c) form a triangle with area A = 1/2 |bc - ad|
- By Pick's theorem, Area = I + B/2 - 1; for unimodular Farey neighbors, bc - ad = 1, giving Area = 1/2
- An area of 1/2 means the triangle has no interior lattice points (I = 0) and exactly 3 boundary points (B = 3), making it an elementary lattice triangle
- The Euclidean distance between centers equals the sum of radii if and only if |bc - ad| = 1
- All Ford circles lie in the upper half-plane tangent to the line y = 0 at (p/q, 0) and have pairwise disjoint interiors
- The Farey adjacency relation bc - ad = 1 implies -ad ≡ 1 (mod b), or ad ≡ -1 (mod b)
- Solving for d using the modular inverse of a modulo b identifies the possible denominators of the neighbor
- In F_n, the adjacent neighbor maximizes the denominator d such that n - b < d ≤ n, or uniquely satisfies d ≤ n (for 3/7 in F_7, 3d ≡ -1 mod 7 gives d ≡ 2 mod 7, so d = 2, yielding c/d = 1/2)
- The unimodular invariant bc - ad = 1 holds for initial neighbors 0/1 and 1/0 and is strictly preserved when inserting mediant (a+c)/(b+d) between a/b and c/d
- By Bézout's identity, bc - ad = 1 implies gcd(a+c, b+d) = 1, ensuring every generated mediant fraction is inherently in reduced form
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 Hidden Structure of Fractions 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 is the mediant of 3/7 and 4/9?
Get it right to open this lesson and 11 more in the app.
How to learn The Hidden Structure of Fractions, step by step
Go in order. 12 short levels across 2 sections, about 24 minutes in all, five minutes a day.
- 1
Navigating the Rational Landscape
- Farey Sequences and the Mediant
- The Stern-Brocot Tree and Rational Addresses
- 2
Approximation and Decomposition
- Continued Fractions and Best Approximations
- Unit Fractions and Greedy Partitions
2 sections · 4 units · 12 levels. Built to play, not to enroll.
Questions about The Hidden Structure of Fractions
How do I learn The Hidden Structure of Fractions?
Go in order: start with Navigating the Rational Landscape, then Approximation and Decomposition. On Tomo that's 12 short levels, about 24 minutes in all, played five minutes a day. Start free and play the first level in under a minute.
Is there a Duolingo-style app for The Hidden Structure of Fractions?
Yes. Tomo works like Duolingo: short daily lessons, streaks and quick quizzes. The Hidden Structure of Fractions is one of its free courses, and you can type any other topic to get a course of your own.
Is The Hidden Structure of Fractions free?
Yes. The Hidden Structure of Fractions is playable on Tomo's free plan, which doesn't expire. No credit card, no waitlist. Download the app and start your first lesson.
How long does it take to learn The Hidden Structure of Fractions?
12 levels across 2 sections, about 24 minutes of play in total. It's built for five minutes a day, so you finish it a little at a time instead of all at once.
Do I need any background to start The Hidden Structure of Fractions?
No. It's pitched at an advanced level, so it starts where you are and builds up. If you can tap a phone, you can start.
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