Chapter P: Prerequisite Information

1. Chapter P:Prerequisite Information Section P-1: Real Numbers

2. Objectives: • You will learn about: • Representing real numbers • Order and interval notation • Basic properties of algebra • Integer exponents • Scientific notation • Why: • These topics are fundamental in the study of mathematics and science.

3. Vocabulary • Real number • Natural (counting) number • Whole number • Integer • Elements (objects) • Rational number • Set builder notation • Terminates • Infinitely repeating • Irrational • Real number line • Origin • Positive numbers • Negative numbers • The coordinate of a point • Ordered • Inequality symbols • closed. • Bounded intervals • Endpoints • Open • Unbounded interval • Variable • Constant • Algebraic expression • Subtraction • Division • Additive inverse (opposite) • Multiplicative inverse (reciprocal) • Factored form • Expanded form • Exponent • Base • nth power of a • Scientific notation • Magnitude

4. Representing Real Numbers • A real number is any number that can be written as a decimal. • Real numbers are either • Rational Numbers-rational numbers can be written as ratios • Integers • Whole numbers • Natural numbers • Irrational Numbers-numbers that are non-rational and infinitely repeating

5. Example 1:Examining Decimal Forms of Rational Numbers • Determine the decimal form of: • 1/16 • 55/27 • 1/17

6. Representing Real Numbers (continued) • The real numbers and the points of a line can be matched one-to-one to form a real number line. • The real number 0 is matched with the origin. • Positive numbers are to the right of the origin. • Negative numbers are to the left of the origin. • The number associated with each point is the coordinate of the point.

7. Order and Interval Notation • The set of real numbers is ordered. • We can compare any two numbers that are not equal using inequality symbols: • > • < • ≥ • ≤

8. Order of Real Numbers

9. Trichotomy Property • Let a and b be any real numbers. • Exactly one of the following is true: • a < b • a = b • a > b

10. Example 2:Interpreting Inequalities • Describe and graph the interval of real numbers for the inequality: • x < 3 • - 1 < x ≤ 4

11. Example 3:Writing Inequalities • Write an interval of real numbers using an inequality and draw its graph. • The real numbers between -4 and -0.5 • The real numbers greater than or equal to 0.

12. Bounded Intervals of Real Numbers:Let a and b be real numbers, a < b

13. Unbounded Intervals of Real Numbers:Let a and b be real numbers, a < b

14. Example 4:Converting Between Intervals and Inequalities • For each example: • Convert interval notation to inequality notation or vice versa. • Find the endpoints and state whether the interval is bounded and state its type. • Graph the interval. • Examples: • [-6, 3) • (-∞, -1) • -2 ≤ x ≤ 3

15. Properties of Algebra • Commutative Property • Associative Property • Identity Property • Inverse Property • Distributive Property

16. Example 5:Using the Distributive Property • Write the expanded form of a (x + 2). • Write the factored form of 3y-by

17. Properties of the Additive Inverse • Let u, v, be real numbers, variables, or algebraic expressions. • Properties: • - (-u) = u • (-u)v = -(uv) • (-u)(-v) = uv • (-1)u = -u • -(u + v) = - u - v

18. Exponential Notation • Let: • a be a real number, variable, or algebraic expression. • n be a positive integer • Then: • an = a∙a∙……∙a • Where: • a is the base • n is the exponent • an is the nth power of a.

19. Example 6:Identifying the Base • (-3)5 • -35

20. Properties of Exponents • Let: • u and v be real numbers, variables, or algebraic expressions. • m, n be integers • All bases are assumed to be nonzero

21. Example 7:Simplifying Expressions Involving Powers • Simplify the following expressions:

22. Example 8:Converting to and from Scientific Notation • 2.375 × 108 = ___________ • 0.000000349

23. Example 9:Using Scientific Notation • Simplify: