Pre-Algebra

An exponent tells you how many times a number (the base) is multiplied by itself. Writing 2⁵ instead of 2×2×2×2×2 is a compact notation that becomes essential once numbers get large, and it obeys its own set of algebraic rules for combining, multiplying, and dividing powers. Exponents extend naturally beyond positive whole numbers to zero, negative, and even fractional values, each with a consistent meaning that preserves the same rules.

Scientific notation writes a number as a decimal between 1 and 10 multiplied by a power of 10, making it compact to write and compare extremely large or small numbers — the mass of the sun (about 1.989 × 10³⁰ kg) or the size of an atom (about 1 × 10⁻¹⁰ m) would otherwise require writing out dozens of digits.

A variable is a symbol, usually a letter like x, y, or n, that stands in for a number whose value is unknown, unspecified, or free to vary. An expression combines variables, numbers, and operations (like +, −, ×, ÷) into a single mathematical phrase — but unlike an equation, it makes no claim of equality and cannot be 'solved,' only simplified or evaluated. Variables and expressions are the basic vocabulary of algebra: they let us write general statements about numbers instead of working with one specific number at a time.

A factor of a whole number n is a whole number that divides n exactly, with no remainder. A multiple of n is any number obtained by multiplying n by a whole number. These two ideas are mirror images of each other: 4 is a factor of 12 because 12 ÷ 4 has no remainder, and 12 is a multiple of 4 because 12 = 4 × 3. Factors and multiples underlie fraction simplification, finding common denominators, and scheduling problems (like figuring out when two repeating events coincide).

Prime factorization is the process of breaking a whole number down into the unique set of prime numbers that multiply together to produce it. A prime number is a whole number greater than 1 whose only factors are 1 and itself (2, 3, 5, 7, 11, ...). Every whole number greater than 1 has exactly one prime factorization (up to the order of the factors) — a fact so fundamental it's called the Fundamental Theorem of Arithmetic — and this decomposition is the key tool for finding GCFs, LCMs, and simplifying fractions systematically.

A negative number is a number less than zero, written with a minus sign (e.g. -3), representing a quantity in the opposite direction or sense from a positive value — debt instead of credit, below zero instead of above, leftward instead of rightward. Negative numbers, together with zero and the positive numbers, form the integers, and they follow consistent rules for addition, subtraction, and multiplication that extend ordinary arithmetic.

The number line is a straight line on which every point corresponds to a real number, with numbers increasing from left to right. It gives a visual, geometric picture of order (which number is bigger), distance (how far apart two numbers are), and operations like addition (moving right) and subtraction (moving left).

Rounding replaces a number with a nearby, simpler value at a chosen level of precision (e.g. to the nearest ten, or nearest hundredth), while estimation uses rounded or approximate values to quickly get a reasonable idea of an answer without computing it exactly. Both are essential practical skills — for checking whether an exact calculation is plausible, and for situations where perfect precision isn't needed or available.

A percentage is a ratio expressed as a fraction of 100, denoted with the % symbol — 'percent' literally means 'per hundred.' Percentages give a standard scale for comparing proportions (a 25% discount means the same relative reduction whether applied to $4 or $4,000), which is why they're used everywhere from sales tax to test scores to statistics.

A ratio compares two quantities by division, showing their relative sizes — for example, a recipe might call for flour and sugar in a ratio of 3 to 1. A proportion is a statement that two ratios are equal, and it's the tool used to scale quantities up or down consistently: if you know 3 apples cost $2, a proportion tells you how much 12 apples cost. Together, ratios and proportions are the backbone of scaling, unit conversion, map reading, and mixing.