Numbers and Number Sets

Types of Numbers

Understanding the different types of numbers is foundational in precalculus. Numbers are classified into several sets, each with unique properties.

• Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...).

• Whole Numbers: Natural numbers plus zero (0, 1, 2, 3, ...).

• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).

• Rational Numbers: Numbers that can be expressed as a fraction , where and are integers and .

• Irrational Numbers: Numbers that cannot be expressed as a fraction, such as or .

• Real Numbers: All rational and irrational numbers.

Example: is rational; is irrational.

Prime Factorization and Least Common Multiple (LCM)

Prime Factorization

Prime factorization is expressing a number as a product of its prime factors.

• Prime Number: A number greater than 1 with only two positive divisors: 1 and itself.

• Example: The prime factorization of 156 is .

Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest number that is a multiple of each.

• Example: LCM of 8, 24, and 40 is 120.

Algebraic Expressions and Operations

Evaluating and Simplifying Expressions

Algebraic expressions can be simplified using properties of arithmetic and exponents.

• Order of Operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

• Example: is a simplified form of .

Factoring Polynomials

Factoring is expressing a polynomial as a product of its factors.

• Common Methods: Factoring out the greatest common factor (GCF), grouping, difference of squares, trinomials.

• Example:

Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.

• Example:

Radicals and Rationalizing the Denominator

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots.

• Example:

Rationalizing the Denominator

Rationalizing the denominator means rewriting a fraction so that the denominator contains no radicals.

• Method: Multiply numerator and denominator by a suitable radical to eliminate the radical in the denominator.

• Example:

Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, typically in the form .

• Example: Solve by factoring: or

Solving for a Variable

Isolating a variable in an equation involves algebraic manipulation.

• Example: Solve for in

Solving Word Problems

Word problems require translating real-world scenarios into mathematical equations.

• Example: Two cars travel in opposite directions. If one travels 5 mph faster than the other and they are 267 miles apart after 3 hours, set up equations to solve for their speeds.

Functions and Graphs

Graphing Linear Equations

Linear equations can be graphed on the coordinate plane. The general form is .

• Slope (): Measures the steepness of the line.

• Y-intercept (): The point where the line crosses the y-axis.

• Example: Graph by solving for and plotting points.

Finding the Equation of a Line

Given two points, the equation of the line can be found using the point-slope form.

• Formula: , where

• Example: Find the equation of the line through and .

Properties of Exponents and Radicals

Exponent Rules

Exponents follow specific rules for multiplication, division, and powers.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Negative Exponent:

Radical and Exponential Notation

Radicals can be expressed in exponential form:

• Example:

Factoring and Simplifying Complex Fractions

Factoring Techniques

Factoring is used to simplify expressions and solve equations.

• Difference of Squares:

• Trinomials: can often be factored into

Complex Fractions

Complex fractions have fractions in the numerator, denominator, or both. Simplify by finding a common denominator.

• Example:

Summary Table: Key Properties and Operations

Additional info:

• Some questions involve graphing and interpreting linear equations, which is a core skill in precalculus.

• Problems on rationalizing denominators and simplifying radicals reinforce understanding of exponents and roots.

• Factoring polynomials and solving equations are essential for progressing to higher-level algebra and calculus.