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Real Numbers and Introduction to Algebra

Chapter 1. Real Numbers and Introduction to Algebra. Chapter Sections. 1.1 – Tips for Success in Mathematics 1.2 – Symbols and Sets of Numbers 1.3 – Exponents, Order of Operations, and Variable Expressions 1.4 – Adding Real Numbers 1.5 – Subtracting Real Numbers

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Real Numbers and Introduction to Algebra

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  1. Chapter 1 Real Numbers and Introduction to Algebra

  2. Chapter Sections 1.1 – Tips for Success in Mathematics 1.2 – Symbols and Sets of Numbers 1.3 – Exponents, Order of Operations, and Variable Expressions 1.4 – Adding Real Numbers 1.5 – Subtracting Real Numbers 1.6 – Multiplying and Dividing Real Numbers 1.7 – Properties of Real Numbers 1.8 – Simplifying Expressions

  3. Tips for Success in Mathematics § 1.1

  4. Getting Ready for This Course

  5. General Tips for Success Continued

  6. General Tips for Success

  7. Using This Text Continued

  8. Using This Text

  9. Getting Help

  10. Preparing for and Taking an Exam Continued

  11. Preparing for and Taking an Exam

  12. Managing Your Time

  13. Symbols and Sets of Numbers § 1.2

  14. Set of Numbers • Natural numbers– {1, 2, 3, 4, 5, 6 . . .} • Whole numbers– {0, 1, 2, 3, 4 . . .} • Integers– {. . . –3, -2, -1, 0, 1, 2, 3 . . .} • Rationalnumbers– the set of all numbers that can be expressed as a quotient of integers, with denominator  0 • Irrationalnumbers– the set of all numbers that can NOT be expressed as a quotient of integers • Realnumbers– the set of all rational and irrational numbers combined

  15. Equality and Inequality Symbols

  16. The Number Line – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 – 4.8 1.5 Negative numbers Positive numbers Anumber lineis a line on which each point is associated with a number.

  17. Order Property for Real Numbers Example For any two real numbers a and b, a is less than b if a is to the left of b on the number line. • a < b means a is to the left of b on a number line. • a > b means a is to the right of b on a number line. Insert < or > between the following pair of numbers to make a true statement.

  18. Absolute Value – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 Symbol for absolute value Distance of 4 Distance of 5 The absolute value of a real number a, denoted by |a|, is the distance between a and 0 on the number line. | – 4| = 4 |5| = 5

  19. Exponents, Order of Operations, and Variable Expressions § 1.3

  20. Example can be written as Expression In Words “three to the second power” or “three squared.” “three to the third power” or “three cubed” “three to the fourth power” Using Exponential Notation We may use exponential notation to write products in a more compact form. Evaluate 26.

  21. Using the Order of Operations

  22. Example Using the Order of Operations Evaluate: Write 32 as 9. Divide 9 by 3. Add 3 to 6. Divide 9 by 9.

  23. Evaluating Algebraic Expressions

  24. Determining Whether a Number is a Solution – 7 is not a solution.

  25. Translating Phrases

  26. Translating Phrases Example Write as an algebraic expression. Use x to represent “a number.” a.) 5 decreased by a number b.) The quotient of a number and 12 a.) In words: 5 decreased by a number – Translate: 5 x The quotient of b.) In words: a number and 12 Translate:  x 12

  27. Adding Real Numbers § 1.4

  28. Adding Real Numbers Adding 2 numbers with the same sign • Add their absolute values. • Use common sign as sign of sum. Adding 2 numbers with different signs • Take difference of absolute values (smaller subtracted from larger). • Use the sign of larger absolute value as sign of sum.

  29. Additive Inverses Example Opposites or additive inversesare two numbers the same distance from 0 on the number line, but on opposite sides of 0. The sum of a number and its opposite is 0. If a is a number, – (– a) = a. Add the following numbers. (–3) + 6 + (–5) = –2

  30. Subtracting Real Numbers § 1.5

  31. Subtracting Real Numbers Example Subtracting real numbers • Substitute the opposite of the number being subtracted • Add. • a – b = a + (– b) Subtract the following numbers. (– 5) – 6 – (– 3) = (– 5) + (– 6) + 3 = – 8

  32. Complementary Angles Example x 150 – 2x Complementary anglesare two angles whose sum is 90o. Find the measure of the following complementary angles. x + 150 – 2x = 90 150 – x = 90 – x = – 60 x = 60° and 150 – 2x = 30°

  33. Supplementary Angles Example x x + 78 Supplementary anglesare two angles whose sum is 180o. Find the measure of the following supplementary angles. x + x + 78 = 180 2x + 78 = 180 2x = 102 x = 51° and x + 78 = 129°

  34. Multiplying and Dividing Real Numbers § 1.6

  35. Multiplying or Dividing Real Numbers Multiplying or dividing 2 real numbers with same sign • Result is a positive number Multiplying or dividing 2 real numbers with different signs • Result is a negative number

  36. Multiplying or Dividing Real Numbers Example Find each of the following products. 4 · (–2) · 3 = –24 (–4) · (–5) = 20

  37. Multiplicative Inverses (Reciprocals) a 0 If b is a real number, 0 · b = b · 0 = 0. Multiplicative inverses or reciprocals are two numbers whose product is 1. The quotient of any real number and 0 is undefined. The quotient of 0 and any real number = 0.

  38. Simplifying Real Numbers Example If a and b are real numbers, and b  0, Simplify the following.

  39. Properties of Real Numbers § 1.7

  40. Commutative and Associative Property • Commutative property • of addition: a + b = b + a • of multiplication: a · b = b ·a • Associative property • of addition: (a + b) + c = a + (b + c) • of multiplication: (a · b) · c = a · (b · c)

  41. Distributive Property • Distributive property of multiplication over addition • a(b + c) = ab + ac • Identities • for addition: 0 is the identity since a + 0 = a and 0 + a = a. • for multiplication: 1 is the identity since a· 1 = a and 1 · a = a.

  42. Inverses • For multiplication: b and are inverses (b  0) • since b · = 1. • Inverses • For addition: a and –a are inverses since a + (– a) = 0.

  43. Simplifying Expressions § 1.8

  44. Terms A term is a number, or the product of a number and variables raised to powers – (the number is called a coefficient) Examples of Terms 7 (coefficient is 7) 5x3(coefficient is 5) 4xy2(coefficient is 4) z2(coefficient is 1)

  45. Like Terms You can combine like terms by adding or subtracting them (this is not true for unlike terms) Like terms contain the same variables raised to the same powers To combine like terms, add or subtract the numerical coefficients (as appropriate), then multiply the result by the common variable factors

  46. Combining Like Terms 6x2 + 7x2 19xy – 30xy 13xy2 – 7x2y 13x2 -11xy Can’t be combined (since the terms are not like terms) Examples of Combining Terms Terms Before Combining After Combining Terms

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