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Activator. 1. Evaluate y^2 / ( 3ab + 2) if y = 4; a = -2; and b = -5 2. Find the value: √17 = 0.25 x 0 = 6 : 10 =. Introduction to Algebra 2 Real Numbers and Their Properties. CCSS: N.RN.3.

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  1. Activator • 1. Evaluate • y^2 / ( 3ab + 2) if y = 4; a = -2; and b = -5 2. Find the value: √17 = 0.25 x 0 = 6 : 10 =

  2. Introduction to Algebra 2Real Numbers and Their Properties CCSS: N.RN.3

  3. Domain:   The Real Number System • Clusters:  Use properties of rational and irrational numbers • Standards: N.RN.3 EXPLAIN why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. N.RN.3

  4. A set is a collection of objects called the elements or members of the set. Set braces { } are usually used to enclose the elements. In Algebra, the elements of a set are usually numbers. • Example 1: 3 is an element of the set {1,2,3} Note: This is referred to as a Finite Setsince we can count the elements of the set. • Example 2: N= {1,2,3,4,…} is referred to as a Natural Numbers or Counting Numbers Set. • Example 3: W= {0,1,2,3,4,…} is referred to as a Whole Number Set. Write sets using set notation

  5. A set is a collection of objects called the elements or members of the set. Set braces { } are usually used to enclose the elements. • Example 4: A set containing no numbers is shown as { } Note: This is referred to as the Null Set or Empty Set. Caution: Do not write the {0} set as the null set. This set contains one element, the number 0. Example 5: To show that 3 “is a element of” the set {1,2,3}, use the notation: 3  {1,2,3}. Note: This is also true: 3N Example 6: 0  N where  is read as “is not an element of set” Write sets using set notation

  6. Two sets are equal if they contain exactly the same elements. (Order doesn’t matter) • Example 1: {1,12} = {12,1} • Example 2: {0,1,3}  {0,2,3} Write sets using set notation

  7. In Algebra, letters called variables are often used to represent numbers or to define sets of numbers. (x or y). The notation {x|x has property P}is an example of “Set Builder Notation” and is read as: {x x has property P} the set of all elements x such that x has a property P • Example 1: {x|x is a whole number less than 6} Solution:{0,1,2,3,4,5} Write sets using set notation

  8. In Algebra, letters called variables are often used to represent numbers or to define sets of numbers. (x or y). The notation {x|x has property P}is an example of “Set Builder Notation” and is read as: {x x has property P} the set of all elements x such that x has a property P • Example 2: {x|x is a natural number greater than 12} Solution:{13,14,15,…} Write sets using set notation

  9. -2 -1 0 1 2 3 4 5 • One way to visualize a set a numbers is to use a “Number Line”. • Example 1: The set of numbers shown above includes positive numbers, negative numbers and 0. This set is part of the set of “Integers” and is written: I = {…, -2, -1, 0, 1, 2, …} Using a number line

  10. Graph of -1 -2 -1 0 1 2 3 4 5 o o o coordinate • Each number on a number line is called the coordinate of the point that it labels, while the point is the graph of the number. • Example 1: The fractions shown above are examples of rational numbers. A rational number is one than can be expressed as the quotient of two integers, with the denominator not 0. Using a number line

  11. o o o o o Graph of -1 -2 -1 0 1 2 3 4 5 o o o coordinate • Decimal numbers that neither terminate nor repeat are called “irrational numbers”. • Example 1: Many square roots are irrational numbers, however some square roots are rational. • Irrational: Rational: Using a number line

  12. The following sets of numbers will be used for this course: • N - Natural numbers or Counting Numbers: {1, 2, 3, …} • W – Whole numbers: {0, 1, 2, 3, …} • I – Integer numbers: {…,-2, -1, 0, 1, 2, …} • R - Rational numbers: • IR – Irrational numbers: {x| x is a real number that is not rational} • RN- Real numbers: {x| x is represented by a point on a number line} The common sets of numbers

  13. Real Numbers The Real Number System Rational Numbers Irrational Numbers 1/2 -2 2 3 3 2/3 0 • 5 • 2 3 4 15% -0.7 1.456

  14. Real Numbers The Real Number System Rational Numbers Irrational Numbers Integers 2 3 3 -2 1/2 2/3 0 • 5 • 2 3 4 - 0.7 15% 1.456

  15. Real Numbers The Real Number System Rational Numbers Irrational Numbers Integers Whole 2 3 1/2 3 2/3 0 • 5 • 2 3 4 -2 - 0.7 15% 1.456

  16. Real Numbers The Real Number System Rational Numbers Irrational Numbers Integers Whole 2 Natural 3 1/2 2/3 0 3 • 5 • 2 3 4 -2 - 0.7 15% 1.456

  17. Question: Select all the words from the following list that apply to the number: Whole Number, Rational Number, Irrational Number, Real Number, Undefined Solution:Whole Number, Rational Number, Real Number The common sets of numbers

  18. For any real number x, the number –x is the additive inverse of x. Example 1: Finding Additive inverses

  19. Examples: Find : To find the absolute value of a signed number: Caution: The absolute value of a number is always positive Using the Absolute Value

  20. Equality/Inequality Symbols: Caution: With the symbol, if either the  or the = part is true, then the inequality is true. This is also the case for the  symbol. Using Inequality Symbols

  21. ) -2 -1 0 1 2 3 4 5 [ -2 -1 0 1 2 3 4 5 • A parenthesis ( or ) is used to indicate a number isnot an element of a set. A bracket [ or ] is used to indicate a number is a member of a set. Example 1: Write in interval notation and graph: {x|x 3} Solution: Interval Notation (-,3) Example 2: Write in interval notation and graph: {x|x 0} Solution: Interval Notation [0, ) Graphing Sets of Real Numbers

  22. [ ) -2 -1 0 1 2 3 4 5 • A parenthesis ( or ) is used to indicate a number isnot an element of a set. A bracket [ or ] is used to indicate a number is a member of a set. Example 3: Write in interval notation and graph: {x| -2  x  3} Solution: Interval Notation [-2,3) Graphing Sets of Real Numbers

  23. Example: Add (–5.6) + (-2.1) = • Example: To add signed numbers: • If the numbers are alike, add their absolute values and use the common sign. • If the numbers are not alike, subtract the smaller absolute value from the larger absolute value. Use the sign of the larger absolute value. Adding Real Numbers

  24. Example: Subtract (–56) - (-70) = • Example: To subtract signed numbers: • Rewrite as an addition problem by adding the opposite of the number to be subtracted. Find the sum. Subtracting Real Numbers

  25. o o -2 -1 0 1 2 3 4 5 • To find the distance between two points on a number line, find the absolute value of the difference between the two points. Example 1: Find the distance between the points: –2 and 3 Solution: |(-2) – (3)| = |-2 -3| = |-5| = 5 or|(3) – (-2)| = |3 +2| = |5| =5 Finding the distance between two points on a number line.

  26. To find the product of a positive and negative signed number: Find the product of the absolute values. Make the sign negative. • Example: Multiply (–12)(5) = • Example: To find the product of two negative signed numbers: Find the product of the absolute values. Make the sign positive. Multiplying Real Numbers

  27. To find the quotient of a positive and negative signed number: Find the quotient of the absolute values. Make the sign negative. • Example: Divide (–12)(5) = • Example: To find the quotient of two negative signed numbers: Find the quotient of the absolute values. Make the sign positive. Caution: Division by zero is “undefined”: Dividing Real Numbers

  28. To find the Inverse of a real number: Find the reciprocal of the number. Keep the same sign. Caution: A number and its “reciprocal” always have the same sign. A number and its “additive inverse” have opposite signs. Examples: Dividing Real Numbers

  29. Using Exponents • If “a” is a real number and “n” is a natural number, then an = a•a•a•••a•a (n factors of a). where n is the exponent, a is the base, and an is an exponential expression. Exponents are also called powers. • To find the value of a whole number exponent: • 100 = 1, 20 = 1, 80 = 1, #0 = 1 • 101 = 10, 21 = 2, 81 = 8, #1 = # • 102 = 10 x 10 = 100, 22 = 2 x 2 = 4, 82 = 8 x 8 = 64 • 103 = 10 x 10 x 10 = 1000, 23 = 2 x 2 x 2 = 8 • 104 = 10 x 10 x 10 x 10 = 10,000 24 = 2 x 2 x 2 x 2 = 16 • (-10)3 = (-10)(-10)(-10) (12).5 =

  30. Order of Operations • To evaluate an expression: 12 - 9 ÷ 3 = 12 – 3 = 9 (12 – 9) ÷ 3 = (3) ÷ 3 = 1 (43 – 120 ÷ 2)2 + 82 = (64 – 60)2 + 64 = 42 + 64 = 16 + 64 = 80

  31. Order of Operations • Example 1: Simplify Solution: Example 2: Simplify

  32. Evaluating Algebraic Expressions • Example 1: if w = 4, x = -12, y = 64, z = -3 • Find: Solution: Example 2: Find:

  33. For any real numbers a, b , and c a(b + c) = ab + ac or (b + c)a = ab +ac Note: This is often referred to as “removing parenthesis” • Example 1: -4(p – 5) = -4p – 20 • Example 2: -6m +2m = m(-6 + 2) = m(-4) = -4m Note: This is often referred to as “factoring out m” Using the Distributive Property

  34. Zero is the only number that can be added to any number to get that number. 0 is called the “identity element for addition” a + 0 = a Example 1: 4 + 0 = 4 Note: This is referred to as the “additive identity” • One is the only number that can be multiplied by any number to get that number. 1 is called the “identity element for multiplication” a • 1 = a Example 2: 4 • 1 = 4 Note: This is referred to as the “multiplicative identity” Using the Identity Properties

  35. For any real numbers a, b, and c, a + b = b + a and ab = ba Note: These are referred to as the “commutative properties” • For any real numbers a, b, and c, a + (b + c) = (a + b) + c and a(bc) = (ab)c Note: These are referred to as the “associative properties” Using the Commutative and Associate Properties

  36. Example 1: Simplify 12b – 9 + 4b – 7 b +1 = Solution: 12b + 4b – 7b + (-9 + 1) = b(12 +4 -7) + (-8) = 9b - 8 • Example 2: Simplify 6 – (2x + 7) –3 = Solution: -2x -7 + 6 – 3 = -2x -4 Using the Properties of Real Numbers

  37. For any real number a: a • 0 = 0 • Example 1: Simplify (6 – (2x + 7) –3)(0) = Solution: 0 Using the Multiplication Property of 0

  38. Do you understand that you are responsible for your own success or failure? Learning in the 21st Century

  39. Questions ????? Learning in the 21st Century

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