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REAL NUMBER SYSTEM

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Number Systems Real Rational (fraction) Irrational Integer Whole Natural

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Real Number System

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Real Numbers Real numbers consist of all the rational and irrational numbers. The real number system has many subsets: – Natural Numbers – Whole Numbers – Integers

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Natural Numbers Natural numbers are the set of counting numbers. {1, 2, 3,…}

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Whole Numbers Whole numbers are the set of numbers that include 0 plus the set of natural numbers. {0, 1, 2, 3, 4, 5,…}

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Integers Integers are the set of whole numbers and their opposites. {…,-3, -2, -1, 0, 1, 2, 3,…}

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Rational Numbers Rational numbers are any numbers that can be expressed in the form of, where a and b are integers, and b ≠ 0. They can always be expressed by using terminating decimals or repeating decimals.

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Terminating Decimals Terminating decimals are decimals that contain a finite number of digits. Examples: 36.8 0.125 4.5

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Repeating Decimals Repeating decimals are decimals that contain a infinite number of digits. Examples: 0.333… 7.689689… The line above the decimals indicate that number repeats.

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Irrational Numbers Irrational numbers are any numbers that cannot be expressed as They are expressed as non-terminating, non-repeating decimals; decimals that go on forever without repeating a pattern. Examples of irrational numbers: – 0.34334333433334… – 45.86745893… – (pi) –

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Set of Real Numbers Rational Numbers –the set of all numbers that can be expressed as a quotient of integers ie. In p/q form where q is not equal to zero Irrational Numbers – the set of all numbers that correspond to points on the number line but that are not rational numbers. That is, an irrational number is a number that cannot be expressed as a quotient of integers. Real Numbers – is the set of all numbers each of which correspond to a point on the number line.

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Other Vocabulary Associated with the Real Number System …(ellipsis)—continues without end { } (set)—a collection of objects or numbers. Sets are notated by using braces { }. Finite—having bounds; limited Infinite—having no boundaries or limits Venn diagram—a diagram consisting of circles or squares to show relationships of a set of data.

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Venn Diagram of the Real Number System Irrational NumbersRational Numbers

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Example Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that apply. a.117 b.0 c.-12.64039… d.-½ e.6.36 f. g.-3

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To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn diagram. Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers -12.64039… 117 0 6.36 -3

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Solution 117 is a natural number, a whole number, an integer, and a rational number. is a rational number. 0 is a whole number, an integer, and a rational number. -12.64039… is an irrational number. -3 is an integer and a rational number. 6.36 is a rational number. is an irrational number. is a rational number.

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FYI…For Your Information When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non- repeating. Therefore, those numbers are always irrational.

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Example 2-1b Name all of the sets of numbers to which each real number belongs. a. 31 b. c. 4.375 d. e. Answer:natural number, whole number, integer, rational number Answer:integer, rational number Answer:rational number Answer:natural number, whole number, integer, rational number Answer:irrational number The Real Number System (rational and irrational)

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Commutative Properties Changing the order of the numbers in addition or multiplication will not change the result. Commutative Property of Addition states: 2 + 3 = 3 + 2 or a + b = b + a. Commutative Property of Multiplication states: 4 5 = 5 4 or ab = ba.

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Associative Properties Changing the grouping of the numbers in addition or multiplication will not change the result. Associative Property of Addition states: 3 + (4 + 5)= (3 + 4)+ 5 or a + (b + c)= (a + b)+ c Associative Property of Multiplication states: (2 3) 4 = 2 (3 4) or (ab)c = a(bc)

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Distributive Property Multiplication distributes over addition.

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Additive Identity Property There exists a unique number 0 such that zero preserves identities under addition. a + 0 = a and 0 + a = a In other words adding zero to a number does not change its value.

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Multiplicative Identity Property There exists a unique number 1 such that the number 1 preserves identities under multiplication. a ∙ 1 = a and 1 ∙ a = a In other words multiplying a number by 1 does not change the value of the number.

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Additive Inverse Property For each real number a there exists a unique real number –a such that their sum is zero. a + (-a) = 0 In other words opposites add to zero.

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Multiplicative Inverse Property For each real number a there exists a unique real number 1/a such that their product is 1.

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State the property or properties that justify the following. 3 + 2 = 2 + 3 Commutative Property

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State the property or properties that justify the following. 10(1/10) = 1 Multiplicative Inverse Property

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State the property or properties that justify the following. 3(x – 10) = 3x – 30 Distributive Property

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State the property or properties that justify the following. 3 + (4 + 5) = (3 + 4) + 5 Associative Property

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State the property or properties that justify the following. (5 + 2) + 9 = (2 + 5) + 9 Commutative Property

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3 + 7 = 7 + 3 Commutative Property of Addition

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8 + 0 = 8 Identity Property of Addition

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6 4 = 4 6 Commutative Property of Multiplication

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17 + (-17) = 0 Inverse Property of Addition

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(2 + 1) + 4 = 2 + (1 + 4) Associative Property of Addition

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even + even = even Closure Property

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5 1 / 7 + 0 = 5 1 / 7 Identity Property of Addition

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3 / 4 – 6 / 7 = – 6 / 7 + 3 / 4 Commutative Property of Addition

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1 2 / 5 1 = 1 2 / 5 Identity Property of Multiplication

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(fraction)(fraction) = fraction Closure Property

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[(- 2 / 3 )(5)]9 = - 2 / 3 [(5)(9)] Associative Property of Multiplication

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6(3 – 2n) = 18 – 12n Distributive Property

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2x + 3 = 3 + 2x Commutative Property of Addition

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ab = ba Commutative Property of Multiplication

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a + 0 = a Identity Property of Addition

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a(bc) = (ab)c Associative Property of Multiplication

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a1 = a Identity Property of Multiplication

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a +b = b + a Commutative Property of Addition

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a(b + c) = ab + ac Distributive Property

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a + (b + c) = (a +b) + c Associative Property of Addition

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a + (-a) = 0 Inverse Property of Addition

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