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Understanding Real Numbers: Representation, Notation, and Properties

This chapter covers the fundamentals of real numbers, including their representation and order using interval notation. You will learn about the essential properties of algebra, integer exponents, and scientific notation. Understanding these concepts is vital for further studies in mathematics and science. Key vocabulary includes rational numbers, irrational numbers, inequality symbols, and properties of algebra. Engage with examples that demonstrate how to convert between different forms of number notation and explore bounded and unbounded intervals.

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Understanding Real Numbers: Representation, Notation, and Properties

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  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:

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