 # 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 =

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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

N.RN.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.

Write sets using set notation 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 Set since 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 Write sets using set notation 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 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 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 ofall elements xsuch 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 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 ofall elements xsuch that x has a property P Example 2: {x|x is a natural number greater than 12} Solution:{13,14,15,…}

Using a number line  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, …} -2 -1 0 1 2 3 4 5

Using a number line  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. -2 -1 0 1 2 3 4 5 coordinate Graph of -1 ooo

Using a number line  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: -2 -1 0 1 2 3 4 5 coordinate Graph of -1 ooo ooooo

The common sets of numbers  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}

8/5/2015 Real Numbers Rational NumbersIrrational Numbers 3 1/2 -2 15% 2/3 1.456 -0.7 0 33 22 -  5 2 3434

Real Numbers Rational NumbersIrrational Numbers 3 1/2 -2 15% 2/3 1.456- 0.7 0 33 22 -  5 2 3434 Integers

Real Numbers Rational NumbersIrrational Numbers 31/2 -2 15% 2/3 1.456- 0.7 0 33 22 -  5 2 3434 Integers Whole

Real Numbers Rational NumbersIrrational Numbers 3 1/2 -2 15% 2/3 1.456- 0.7 0 33 22 -  5 2 3434 Integers Whole Natural

The common sets of numbers  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

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

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

Using Inequality Symbols  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.

Graphing Sets of Real Numbers  A parenthesis ( or ) is used to indicate a number is not 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,  ) -2 -1 0 1 2 3 4 5 ) [

Graphing Sets of Real Numbers  A parenthesis ( or ) is used to indicate a number is not 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) -2 -1 0 1 2 3 4 5 )[

Adding Real Numbers Adding Real Numbers  Example: Add (–5.6) + (-2.1) =  Example: To add signed numbers: 1) If the numbers are alike, add their absolute values and use the common sign. 2) If the numbers are not alike, subtract the smaller absolute value from the larger absolute value. Use the sign of the larger absolute value.

Subtracting Real Numbers  Example: Subtract (–56) - (-70) =  Example: To subtract signed numbers: 1) Rewrite as an addition problem by adding the opposite of the number to be subtracted. Find the sum.

Finding the distance between two points on a number line.  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 -2 -1 0 1 2 3 4 5 oo

Multiplying Real Numbers 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.

Dividing Real Numbers 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 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:

Using Exponents  If “a” is a real number and “n” is a natural number, then a n = aaaaa (n factors of a). where n is the exponent, a is the base, and a n is an exponential expression. Exponents are also called powers.  To find the value of a whole number exponent: 10 0 = 1, 2 0 = 1, 8 0 = 1, # 0 = 1 10 1 = 10, 2 1 = 2, 8 1 = 8, # 1 = # 10 2 = 10 x 10 = 100, 2 2 = 2 x 2 = 4, 8 2 = 8 x 8 = 64 10 3 = 10 x 10 x 10 = 1000, 2 3 = 2 x 2 x 2 = 8 10 4 = 10 x 10 x 10 x 10 = 10,000 2 4 = 2 x 2 x 2 x 2 = 16 (-10) 3 = (-10)(-10)(-10) (12).5 =

Order of Operations  To evaluate an expression: 12 - 9 ÷ 3 = 12 – 3 = 9 (12 – 9) ÷ 3 = (3) ÷ 3 = 1 (4 3 – 120 ÷ 2) 2 + 8 2 = (64 – 60) 2 + 64 = 4 2 + 64 = 16 + 64 = 80

Order of Operations Example 1: Simplify Solution: Example 2: Simplify

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

Using the Distributive Property  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 Identity Properties  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 Commutative and Associate Properties Using the Commutative and Associate Properties  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 Properties of Real Numbers  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 Multiplication Property of 0  For any real number a: a 0 = 0  Example 1: Simplify (6 – (2x + 7) –3)(0) = Solution: 0

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