1.1 - 1.3

Real Numbers

The set of all rational and irrational numbers. Includes natural numbers, whole numbers, integers, fractions, decimals, and irrational numbers.

Natural Numbers (Counting Numbers)

Positive integers used for counting: 1, 2, 3, 4, …

Whole Numbers

Natural numbers plus zero: 0, 1, 2, 3, …

Integers

Whole numbers and their opposites: …, -3, -2, -1, 0, 1, 2, 3, …

Rational Numbers

Numbers that can be written as a fraction a/b where a and b are integers and b ≠ 0.

Examples of Rational Numbers

3/4, -2, 0.5, 0.333…, 7.

Irrational Numbers

Numbers that cannot be written as a fraction. Their decimals never terminate or repeat.

Examples of Irrational Numbers

π, √2, √3, e.

Real Number Line

Every point on the number line represents a real number.

Terminating Decimal

A decimal that ends (e.g., 0.25).

Repeating Decimal

A decimal with a repeating pattern (e.g., 0.666…).

Is 0 Rational?

Yes, because 0 = 0/1.

Subset Relationship

Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real.

Closure Property

Addition, subtraction, multiplication, and division (except by 0) of rational numbers produce rational numbers.

Commutative Property of Addition

a + b = b + a.

Commutative Property of Multiplication

ab = ba.

Associative Property of Addition

(a + b) + c = a + (b + c).

Associative Property of Multiplication

(ab)c = a(bc).

Distributive Property

a(b + c) = ab + ac.

Absolute Value

The distance from 0 on the number line. Always nonnegative.

Opposite (Additive Inverse)

Two numbers with the same distance from 0 but opposite signs.

Identity Property of Addition

a + 0 = a.

Identity Property of Multiplication

a × 1 = a.

Inverse Property of Addition

a + (-a) = 0.

Multiplicative Inverse

The reciprocal of a number. For a ≠ 0, a × (1/a) = 1.

Prime Number

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

Composite Number

A whole number greater than 1 with more than two positive factors.

Exponent

A small number indicating how many times the base is multiplied by itself.

Base

The repeated factor in an exponential expression.

Power

The entire exponential expression (e.g., 2⁴).

Expanded Form of Exponents

2⁴ = 2 × 2 × 2 × 2.

Exponent of 1

a¹ = a.

Exponent of 0

For a ≠ 0, a⁰ = 1.

Negative Exponent

a⁻ⁿ = 1/aⁿ.

Product of Powers

aᵐ × aⁿ = aᵐ⁺ⁿ.

Quotient of Powers

aᵐ / aⁿ = aᵐ⁻ⁿ (a ≠ 0).

Power of a Power

(aᵐ)ⁿ = aᵐⁿ.

Power of a Product

(ab)ⁿ = aⁿbⁿ.

Power of a Quotient

(a/b)ⁿ = aⁿ/bⁿ (b ≠ 0).

Zero Base

0ⁿ = 0 for n > 0.

Undefined Expression

0⁰ is undefined.

Scientific Notation

A way to write very large or very small numbers as a × 10ⁿ where 1 ≤ |a| < 10.

Coefficient in Scientific Notation

The number a in a × 10ⁿ.

Exponent in Scientific Notation

The power of 10 that shows how many places the decimal moves.

Positive Exponent in Scientific Notation

Means the original number is greater than or equal to 10.

Negative Exponent in Scientific Notation

Means the original number is between 0 and 1.

Convert to Scientific Notation

Move the decimal so one nonzero digit remains to the left; count places moved for the exponent.

Convert from Scientific Notation

Move the decimal right for positive exponents and left for negative exponents.

Example of Scientific Notation

4,500,000 = 4.5 × 10⁶.

Example of Small Scientific Notation

0.00072 = 7.2 × 10⁻⁴.

Significant Digits in Scientific Notation

All digits in the coefficient are significant.

Square Root

A number that, when multiplied by itself, equals the original number.

Cube Root

A number that, when multiplied by itself three times, equals the original number.

Radical Symbol

√

Radicand

The number inside the radical symbol.

Index

The small number indicating the root (usually omitted for square roots).

Perfect Square

A number whose square root is an integer.

Examples of Perfect Squares

1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Perfect Cube

A number whose cube root is an integer.

Examples of Perfect Cubes

1, 8, 27, 64, 125, 216.

Simplifying Radicals

Factor out perfect square factors from the radicand.

Example of Simplifying Radical

√72 = √36 × √2 = 6√2.

Product Property of Radicals

√a × √b = √(ab) (a, b ≥ 0).

Quotient Property of Radicals

√(a/b) = √a / √b (b > 0).

Like Radicals

Radicals with the same index and radicand.

Adding Radicals

Combine only like radicals.

Subtracting Radicals

Subtract coefficients of like radicals.

Multiplying Radicals

Multiply coefficients and radicands separately.

Rationalizing the Denominator

Removing radicals from the denominator by multiplying by an appropriate radical or conjugate.

Conjugates

Expressions of the form a + √b and a − √b.

Difference of Squares

(a + b)(a − b) = a² − b².

Rational Exponent

A fractional exponent representing a root.

Square Root as Exponent

a^(1/2) = √a.

Cube Root as Exponent

a^(1/3) = ∛a.

General Rational Exponent

a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m).

Numerator of a Rational Exponent

Represents the power.

Denominator of a Rational Exponent

Represents the root.

Example of Rational Exponent

8^(2/3) = (∛8)² = 2² = 4.

Converting Radical to Exponent

√x = x^(1/2).

Converting Cube Root to Exponent

∛x = x^(1/3).

Negative Rational Exponent

a^(-m/n) = 1 / a^(m/n).

Domain of Even Roots

Even roots of negative numbers are not real.

Domain of Odd Roots

Odd roots of negative numbers are real.

Simplifying Rational Exponents

Apply exponent rules before evaluating whenever possible.

Order of Operations with Exponents

Evaluate parentheses, exponents, multiplication/division, then addition/subtraction.

Common Exponent Mistake

(a + b)² ≠ a² + b².

Common Radical Mistake

√(a + b) ≠ √a + √b.

Principal Square Root

The nonnegative square root of a number.

Estimate Irrational Roots

Find the two perfect squares the number lies between and estimate the decimal value.

Equivalent Forms of Radicals

√16 = 4 = 16^(1/2).

Equivalent Forms of Cube Roots

∛27 = 3 = 27^(1/3).