Comprehensive Study Notes on Fundamental Arithmetic, Pre-Algebra, and Number Theory

Flashcards defining core concepts, rules, number sets, and arithmetic processes covered in the pre-algebra review notes.

Prime Number

A number greater than $1$ with exactly two factors: $1$ and itself.

Composite Number

A number greater than $1$ that has more than two factors.

Prime Factorization

The process of repeatedly dividing a number by prime numbers until every remaining factor is prime.

Greatest Common Divisor (GCD)

The largest factor shared by both numbers.

Least Common Multiple (LCM)

The smallest positive number into which both numbers divide evenly.

PEMDAS

The order of operations: Parentheses, Exponents, Multiplication/Division from left to right, Addition/Subtraction from left to right.

Reducing a Fraction

Dividing both the numerator and denominator by their greatest common factor.

Multiplying Fractions

Multiplying the numerator by the numerator and the denominator by the denominator, then reducing the result.

Dividing Fractions

Keep, Change, Flip: keep the first fraction, change $\div$ to $\times$, and flip the second fraction.

Adding or Subtracting Fractions

Finding a common denominator before performing the addition or subtraction across the numerators.

Improper Fraction to Mixed Number Conversion

Divide the numerator by the denominator; the quotient becomes the whole number and the remainder is placed over the original denominator.

Fraction to Decimal Conversion

Divide the numerator by the denominator.

Simplifying a Square Root

Find the largest perfect-square factor, split the radical, and take the square root of the perfect square.

Natural Numbers

The set of counting numbers starting at one: $1, 2, 3, 4, \dots$

Whole Numbers

The set of natural numbers including zero: $0, 1, 2, 3, 4, \dots$

Integers

The set of whole numbers and their negative opposites: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$

Rational Number

A number that can be written as a fraction $\frac{a}{b}$ where $b \neq 0$; includes terminating and repeating decimals.

Irrational Number

A real number that cannot be written as a fraction, whose decimal representation does not terminate or repeat.

Multiplying Same Bases Exponent Rule

When multiplying expressions with the same base, add the exponents: $a^m \times a^n = a^{m+n}$.

Dividing Same Bases Exponent Rule

When dividing expressions with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Power-to-a-Power Exponent Rule

When raising a base with an exponent to another power, multiply the exponents: $(a^m)^n = a^{mn}$.

Negative Exponent Rule

An expression with a negative exponent equals its reciprocal with a positive exponent: $a^{-n} = \frac{1}{a^n}$.