Arithmetic
The bedrock of mathematics: what numbers mean, how to combine them, and the small set of rules that govern every later subject. Master these and the rest of math stops feeling like magic.
Counting & Place Value
What a digit actually represents. The base-10 positional system, and why it makes large numbers tractable at all.
Addition & Subtraction
Combining and separating quantities. The commutative and associative properties, carrying, and borrowing.
Multiplication & Division
Repeated addition, sharing, and the inverse pair that turns scaling and splitting into a single mental model.
Order of Operations
Why 2 + 3 × 4 has one right answer and not two. PEMDAS as a tie-breaking convention, not a law of nature.
Negative Numbers
Extending the number line below zero. Signed arithmetic, and the rule that two negatives multiply to a positive — with a reason.
Properties of Operations
Commutative, associative, distributive, identity, and inverse — the small set of laws that make algebraic manipulation legal.
Factors, Multiples & Primes
The hidden structure inside whole numbers. Divisibility, GCD, LCM, and why primes are the atoms.
Fractions
Parts of a whole, written as a ratio. Equivalent fractions, common denominators, and the four operations.
Decimals
Place value extended past the point. Converting between fractions and decimals, and what "repeating" really means.
Percentages
"Per hundred" — a third notation for the same idea as a fraction or decimal. Increases, decreases, and the percent-change trap.
Averages & Mixtures
The arithmetic mean and its weighted cousin, the harmonic and geometric means, and the alligation method for mixture problems.
Profit, Loss & Discount
The merchant's vocabulary — cost price, marked price, selling price — and why two 20% discounts don't add up to 40%.
Interest
Simple and compound interest, the compounding-frequency lever, continuous compounding as a limit, the rule of 72, and partnership splits.
Rate Problems
Distance, work, and flow problems anchored on rate × time = amount. The three canonical templates and the average-speed trap.
Personal Finance
The chapter's payoff. Budgeting, the credit-card minimum-payment trap, amortization, savings goals, inflation, and why starting early matters.
Units & Unit Conversion
Dimensional reasoning and the multiply-by-1 trick. Metric and imperial conversions, compound units, and unit cancellation as a sanity check.