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Learn Mathematical Proofs

Go from computing answers to proving why they are true.

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A mathematical proof is a chain of logical steps showing that a statement must be true, starting from definitions and previously proven results. Learning to write proofs is the main transition from computational math to higher mathematics, and it rests on a small set of techniques: direct proof, proof by contrapositive, proof by contradiction, proof by cases, mathematical induction and disproof by counterexample. Classic first proofs include showing that √2 is irrational and that there are infinitely many primes. Learnr builds a personalized proof-writing course and teaches it in short interactive lessons, with feedback on each step of your reasoning.

How to learn Mathematical Proofs, step by step

Learn the logic underneath proofs. Understand implications, the contrapositive, negation and the quantifiers "for all" and "there exists." Many stuck proofs come from not knowing exactly what the statement claims or what its negation is.
Work from precise definitions. Write the definition of even, divisible, rational or injective before starting each proof, then unpack the hypothesis using it. Proving "the sum of two odd numbers is even" this way is a perfect first exercise.
Practice each technique on classic examples. Prove that √2 is irrational and that there are infinitely many primes by contradiction, and that if n² is even then n is even by contrapositive. Knowing why each technique fits its example helps you choose later.
Get comfortable with induction. Prove sum formulas, divisibility results and inequalities by ordinary induction, then use strong induction for statements like every integer above 1 being a product of primes.
Prove statements about sets and functions. Show two sets are equal by proving each is a subset of the other, and prove functions are injective or surjective from definitions. A proof or two each day in Learnr keeps the techniques fresh.
Write, then rewrite for clarity. Draft on scratch paper, then write a clean version in full sentences that a classmate could follow. Learning LaTeX helps you produce clean proofs, and comparing yours with textbook versions shows what you skipped.

What you'll learn

Read and write statements with quantifiers and implications precisely
Write direct proofs, contrapositive proofs and proofs by contradiction
Prove statements with mathematical induction, including strong induction
Prove facts about sets, functions and integers from definitions
Find counterexamples and write clear, readable proofs

What's inside your Mathematical Proofs course

A sample path Learnr builds for Mathematical Proofs. Your real course adapts to your goal and level.

1. Logic, statements and quantifiers
2. Direct proof and working from definitions
3. Contrapositive, contradiction and cases
4. Mathematical induction
5. Proofs about sets and functions
6. Writing clear proofs and finding counterexamples

How long does it take to learn Mathematical Proofs?

A transition-to-proofs course typically takes about 3 to 4 months at 5 to 7 hours a week. Most people find the first few weeks the hardest, and proof writing keeps improving throughout later courses like real analysis and abstract algebra.

What you need before you start

Solid high school algebra and comfort with functions. Calculus is often taken first for mathematical maturity but is not strictly required.

Common mistakes to avoid

Proving a statement by checking a few examples. Examples can suggest a pattern, but only a general argument proves it for every case.
Assuming what you are trying to prove, or working backward from the conclusion and presenting it as if it were forward reasoning.
Skipping definitions and writing vague steps like "clearly" where the actual reasoning should go.

How Learnr builds your course

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Mathematical Proofs questions, answered

What is a good book for learning proofs?

Popular choices include Daniel Velleman's How to Prove It, which is structured around logic and proof strategies, and Richard Hammack's Book of Proof, which is freely available online from the author.

Why is learning proofs so hard at first?

Earlier math rewards following procedures, while proofs require deciding what to try with no fixed recipe. That skill develops with practice, especially by studying many short proofs closely and writing your own.