Chapter P Prerequisites Course Number

Section P.1 Review of Real Numbers and Their Properties Instructor

Objective: In this lesson you learned how to represent, classify, and Date order real numbers, and to evaluate algebraic expressions.

Important Vocabulary Define each term or concept.

Real numbers The set of numbers formed by joining the set of rational numbers and

the set of irrational numbers.

Inequality A statement that represents an order relationship.

Absolute value The magnitude or distance between the origin and the point

representing a real number on the real number line.

I. Real Numbers (Page 2) What you should learn How to represent and A real number is rational if it can be written as . . . the ratio classify real numbers p/q of two integers, where q ¹ 0. A real number that cannot be written as the ratio of two integers is called irrational. The point 0 on the real number line is the origin . On the number line shown below, the numbers to the left of 0 are negative . The numbers to the right of 0 are positive .

0 Every point on the real number line corresponds to exactly one real number.

Example 1: Give an example of (a) a rational number (b) an irrational number Answers will vary. For example, (a) 1/8 = 0.125 (b) p » 3.1415927 . . .

II. Ordering Real Numbers (Pages 3-4) What you should learn How to order real If a and b are real numbers, a is less than b if b - a is numbers and use positive. inequalities

The symbol < means less than . The symbol > means greater than . The symbol ³ means greater than or equal to . The symbol £ means less than or equal to .

Example 2: Place the correct symbol (< or >) between the 14 numbers: - > - 26 . 3

Inequalities can be used to describe subsets of real numbers called intervals . In the interval [a, b], the real numbers a and b are the endpoints of the interval. The interval (a, b) is called a(n) open interval. Positive infinity, represented by the symbol ¥ , and negative infinity, represented by the symbol -¥ , do not represent real numbers. Instead, these symbols are used to describe the unboundedness of an interval.

Example 3: Write an interval representing the entire real line. (-¥, ¥)

III. Absolute Value and Distance (Page 5) What you should learn How to find the absolute If a is a real number, then the absolute value of a is: values of real numbers a, if a ³ 0 and find the distance ì a = í between two real î - a, if a < 0 numbers

Let a and b be real numbers. The distance between a and b is d(a, b) = | b - a | = | a - b| .

Example 4: Explain how to find the absolute value of a negative number. To find the absolute value of a negative number, remove the negative sign from in front of the number.

IV. Algebraic Expressions (Page 6) What you should learn How to evaluate An algebraic expression is . . . . a collection of letters algebraic expressions (variables) and real numbers (constants) combined using the operations of addition, subtraction, multiplication, division, and exponentiation. The terms of an algebraic expression are those parts that are separated by addition . A term that contains variables is called a variable term, and a term that consists of a number alone is called a constant term. The numerical factor of a variable term is the coefficient of the variable term. To evaluate an algebraic expression, . . . substitute numerical values for each of the variables in the expression.

The Substitution Principle, used when an algebraic expression is evaluated, states that . . . if a = b, then a can be replaced by b in any expression involving a.

Example 5: Use the Substitution Principle to evaluate the algebraic expression 2x + 5 when x = - 2. 1

V. Basic Rules of Algebra (Pages 6-8) What you should learn How to use the basic Define additive inverse. rules and properties of The opposite of a real number. If b is a real number, then – b is algebra its additive inverse.

Define multiplicative inverse. The reciprocal of a real number. If b is a real number, then 1/b is its multiplicative inverse.

Let r, s, and t be real numbers, variables, or algebraic expressions. Use r, s, and t to write an example of each of the following properties: Commutative Property of Multiplication: rs = sr Associative Property of Addition: (r + s) + t = r + (s + t) Distributive Property: r(s + t) = rs + rt Multiplicative Identity Property: t × 1 = t Additive Inverse Property: r + (- r) = 0

List five Properties of Negation. 1) (- 1)a = - a 2) - (- a) = a 3) (- a)b = - (ab) = a(- b) 4) (- a)(- b) = ab 5) - (a + b) = (- a) + (- b)

List four Properties of Equality. 1) If a = b, then a + c = b + c. 2) If a = b, then ac = bc. 3) If a + c = b + c, then a = b. 4) If ac = bc, and c ¹ 0, then a = b.

List five Properties of Zero. 1) a + 0 = a and a - 0 = a 2) a × 0 = 0 3) 0/a = 0, if a ¹ 0 4) a/0 is undefined 5) If ab = 0, then a = 0 or b = 0.

To add or subtract fractions with like denominators, . . . add or subtract the numerators of the fractions and write the result over the like denominator. To multiply two fractions, . . . multiply the numerators and write it over the product of the denominators. Define factor. If a, b, and c are integers such that ab = c, then a and b are factors or divisors of c. A prime number is an integer that . . . has exactly two positive factors: itself and 1. A composite number can be written as . . . the product of two or more prime numbers. The number 1 is neither prime nor composite.

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