Arithmetic Foundations

Arithmetic is the substrate of every course that follows, and fluency in it is less about speed than about structure: knowing how numbers decompose, which operations commute, and what stays invariant under manipulation. This lesson consolidates the structural facts that later lessons will use without comment.

Order of operations and sign discipline

Expressions evaluate in a fixed order: brackets, then exponents, then multiplication and division left to right, then addition and subtraction left to right. Negative numbers obey two rules worth restating precisely: subtracting a negative adds, and a product or quotient of two negatives is positive. Most sign errors come not from ignorance of these rules but from dropping a bracket: while , and the habit of bracketing every negative quantity before manipulating it eliminates the error class entirely.

Divisibility and factorisation

Every whole number factorises uniquely into primes, and the factorisation is the number's skeleton: answers instantly whether 360 is divisible by 12 (yes: fits inside) or by 16 (no: only three 2s available). The standard divisibility tests read the answer off the digits.

Parity and invariants

Parity, evenness or oddness, survives addition in a predictable way: even plus even and odd plus odd are even; even plus odd is odd. Products are even unless every factor is odd. Because parity is preserved by these rules, it works as an invariant: a quantity that cannot change under the allowed operations, used to prove that certain outcomes are impossible. A sum of ten odd numbers is even, whatever the numbers; a sequence of steps that each change a counter by 2 can never move it from even to odd. Arguments from invariants recur throughout mathematics, and parity is the first and most used.