Foundations

The natural numbers are the numbers used for counting (1, 2, 3, ...) and, in most modern conventions, ordering, including 0 as the starting point in set theory and computer science. They are the most basic objects in mathematics — every other number system (integers, rationals, reals, complex numbers) is built by extending the natural numbers to solve equations the natural numbers alone cannot solve.

A real number is any number that measures a continuous quantity — a length, a duration, a temperature. The real numbers include every rational number (fractions like 3/4) and every irrational number (like √2 and π), and together they fill the number line completely, with no gaps. Formally, the reals are the unique complete ordered field: they obey the ordinary rules of arithmetic (a field), come in a consistent order (ordered), and have no missing points (complete).

Mathematical induction is a proof technique for establishing that a statement P(n) holds for every natural number n. It works like an infinite row of dominoes: show the first domino falls (base case), and show that any falling domino knocks over the next one (inductive step). Together those two facts guarantee every domino falls.

A function is a rule that assigns exactly one output to each input from a given set. Functions are the central object of essentially all of mathematics beyond arithmetic — calculus studies how functions change, linear algebra studies a special class of functions (linear transformations), and analysis studies which functions are continuous, differentiable, or integrable.

The absolute value of a real number is its distance from zero on the number line, regardless of direction. Denoted |x|, it strips away the sign of a number: |3| = 3 and |-3| = 3. It's the simplest example of a function that measures magnitude without regard to direction — an idea that generalizes to distance in the plane, norms of vectors, and moduli of complex numbers.

The integers extend the natural numbers by adding zero and the negative whole numbers, forming the set ℤ = {..., -2, -1, 0, 1, 2, ...}. Unlike the natural numbers, the integers are closed under subtraction — for any two integers a and b, a - b is always an integer. This closure is precisely what motivated their invention: solving equations like x + 5 = 3 requires a number system beyond the naturals.

A rational number is any number that can be written as a fraction p/q of two integers, with q ≠ 0. This includes every integer (since n = n/1), every terminating decimal, and every repeating decimal. The rationals fill in the gaps left by the integers — you can always divide two integers and land back inside the same number system — but as the ancient Greeks discovered with √2, the rationals themselves still have gaps, which is exactly what motivates the real numbers.

A fraction represents a part of a whole, written as a numerator over a denominator (a/b), where the denominator tells you how many equal parts the whole is split into and the numerator tells you how many of those parts you have. Fractions are the everyday notation for rational numbers, and mastering their arithmetic — common denominators, simplifying, multiplying, dividing — is the practical foundation the rest of algebra builds on.

The order of operations is the agreed-upon sequence for evaluating a mathematical expression with multiple operations, so that everyone gets the same answer from the same expression. Without a fixed convention, 2 + 3 × 4 could mean either (2+3)×4=20 or 2+(3×4)=14 — the order of operations resolves this ambiguity by fixing which operations happen first.