The real number system is the foundation of algebra. It includes , , , , and . Understanding these categories helps us work with different types of numbers and perform calculations accurately.
Properties of , like commutative and associative properties, are key to simplifying expressions. Knowing how to apply these properties, along with the , allows us to solve complex problems and manipulate algebraic expressions efficiently.
Real Number System
Categories of real numbers
Natural numbers
Positive integers starting from 1 (1, 2, 3, 4)
Used for counting and ordering discrete objects
Whole numbers
Non-negative integers including 0 (0, 1, 2, 3, 4)
Encompasses all natural numbers and zero
Integers
Positive and negative whole numbers, including 0 (-3, -2, -1, 0, 1, 2, 3)
Can represent quantities with opposite directions or values
Rational numbers
Numbers expressed as a ratio of two integers qp, where q=0 (21, 0.5, 0.333..., -2.75)
Includes integers, terminating decimals, and repeating decimals
Can be written as fractions or mixed numbers
Irrational numbers
Real numbers that cannot be expressed as a ratio of two integers (2, π, e)
Have non-repeating, non-terminating decimal expansions
Cannot be written as fractions or mixed numbers
Number Sets and the Real Number Line
Number sets represent collections of numbers with specific properties (e.g., natural numbers, integers, rational numbers)
The is a visual representation of all real numbers
Extends infinitely in both positive and negative directions
Each point on the line corresponds to a unique real number
Absolute value represents the distance of a number from zero on the real number line
Order of operations
PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)
Simplify expressions inside parentheses first
Evaluate exponents and roots
Perform multiplication and division from left to right
Perform addition and subtraction from left to right
Example: 2+3×(42−1)÷5
Parentheses: 2+3×(16−1)÷5
Exponents: 2+3×15÷5
Multiplication and Division (left to right): 2+45÷5
Terms with the same variables and exponents (3x + 2y - x + 4y = 2x + 6y)
Simplifies expressions by reducing the number of terms
Apply distributive property to expand expressions
Multiplies each term inside parentheses by the factor outside (2(3x - 1) = 6x - 2)
Eliminates parentheses and prepares for further simplification
Factor out common terms
Identifies the greatest common factor (GCF) among terms (6x + 9 = 3(2x + 3))
Simplifies expressions and reveals patterns
Simplify fractions
Divide numerator and denominator by their GCF (9x6x2=32x, divide by GCF of 3x)
Reduces fractions to lowest terms for easier manipulation
Solve inequalities
Use algebraic techniques similar to solving equations
Remember to reverse the inequality sign when multiplying or dividing by a negative number
Key Terms to Review (23)
Associative property of multiplication: The associative property of multiplication states that the way numbers are grouped in a multiplication problem does not change the product. Mathematically, for any real numbers $a$, $b$, and $c$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
Base: In mathematics, a base is the number that is raised to a power in an exponential expression. It serves as the foundation for representing numbers in different numeral systems.
Commutative property of addition: The commutative property of addition states that changing the order of addends does not change their sum. Mathematically, it is expressed as $a + b = b + a$.
Commutative property of multiplication: The commutative property of multiplication states that changing the order of the factors does not change the product. Mathematically, for any real numbers $a$ and $b$, $a \times b = b \times a$.
Constant: A constant is a fixed value that does not change. In algebra, constants are often represented by numbers or letters that remain the same throughout an equation or expression.
Distributive property: The distributive property states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products. Mathematically, it is expressed as $a(b + c) = ab + ac$.
Equation: An equation is a mathematical statement that asserts the equality of two expressions. It consists of two expressions separated by an equals sign ($=$).
Exponent: An exponent refers to the number that indicates how many times a base number is multiplied by itself. It is typically written as a small number to the upper right of the base number.
Exponential notation: Exponential notation is a mathematical method for representing numbers as a base raised to an exponent. It simplifies the expression and computation of very large or very small numbers.
Formula: A formula is a mathematical expression that relates variables and constants using algebraic operations. Formulas are used to solve problems by substituting known values into the equation.
Identity property of addition: The identity property of addition states that any number added to zero remains unchanged. Mathematically, for any real number $a$, $a + 0 = a$.
Identity property of multiplication: The identity property of multiplication states that any number multiplied by 1 remains unchanged. Mathematically, for any real number $a$, $a \times 1 = a$.
Integers: Integers are a set of numbers that include all whole numbers and their negative counterparts, including zero. They are represented by the symbol $\mathbb{Z}$.
Inverse property of addition: The inverse property of addition states that for any real number $a$, there exists a number $-a$ such that $a + (-a) = 0$. This property ensures that every real number has an additive inverse.
Inverse property of multiplication: The inverse property of multiplication states that any nonzero number multiplied by its reciprocal equals one. Mathematically, $a \cdot \frac{1}{a} = 1$ for any $a \neq 0$.
Irrational numbers: Irrational numbers are real numbers that cannot be expressed as a simple fraction of two integers. They have non-repeating, non-terminating decimal expansions.
Natural numbers: Natural numbers are the set of positive integers starting from 1 and increasing without bound. They are typically used for counting and ordering.
Order of operations: Order of operations is a set of rules used to evaluate mathematical expressions in a consistent manner. The standard order is Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right) often abbreviated as PEMDAS.
Rational numbers: Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. They can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.
Real number line: The real number line is a straight, continuous line that represents all real numbers ordered from left (negative) to right (positive). Each point on the line corresponds to a unique real number, including integers, fractions, and irrational numbers.
Real numbers: Real numbers include all rational and irrational numbers, encompassing integers, fractions, and non-repeating decimals. They can be represented on a number line.
Variable: A variable is a symbol, usually a letter, that represents one or more numbers. It is used to generalize mathematical statements and expressions.
Whole numbers: Whole numbers are non-negative integers, including zero. They do not include fractions, decimals, or negative numbers.