Notes on Numbers

These notes are an attempt to provide easy-to-follow and concise explanations of some classical results in mathematics, principally in number theory. The chapter titles link to pdfs.

1. Numbers and Division

The naive, geometric interpretation of numbers as points in one or two dimensions; integers, rational, real and algebraic numbers; division and the Division Identity (a basic theorem); and some properties of inequalities.

2. Forms of Proof

Deductive proof; if and only if; necessary and sufficient; the converse and the contrapositive; reductio ad absurdum; infinite descent; and the Principle of Induction.

3. Common Factors and Linear Diophantine Equations

The greatest common divisor of two and more numbers; the Euclidean Algorithm and its efficiency; specific and general solutions to binary linear diophantine equations; the coin problem and Frobenius numbers for the two variable case; specific solutions to multi-variable linear diophantine equations.

4. Prime Numbers and the Unique Factorisation Theorem

The infinity of primes; the Sieve of Eratosthenes; the Prime Number Theorem; the Unique Factorisation Theorem a.k.a. the Fundamental Theorem of Arithmetic; a note on Euclid and prime numbers.

5. The Integers and the Real Numbers

Mappings and Cardinality; the Schroeder-Bernstein Theorem; the rational numbers and the algebraic numbers are countable (can be put into 1:1 correspondence with the positive integers); but the real numbers are not.

6. Binary Quadratic Diophantine Equations
7. Congruence
8. The Prime Number Theorem

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