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The Real Number System Created by Mrs. Gray 2010

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What is the Real Number System? The set of all rational and irrational numbers. { } indicates a set. (braces) All numbers can be classified as rational or irrational.

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FYI……For Your Information …(ellipsis)—continues without end { } (set)—a collection of objects or numbers. Sets are notated by using braces { }. Venn diagram—a diagram consisting of circles or squares to show relationships of a set of data.

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Real Numbers can be classified as: Rational –Fractions (proper, improper and mixed) –Integers (positive and negative numbers) –Whole Numbers –Natural Numbers Irrational

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Natural Numbers Always begin with 1 {1, 2, 3, 4, 5, 6, 7,....} Sometimes referred to as Counting Numbers This is an ellipse Which means it Continues.

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{x | x can be written as a decimal number.} Read as all numbers x, such that x is a decimal. –Examples 3 can be written 3.0 ¼ can be written 0.25 2 ½ can be written as 2.5 -5 can be written as -5.0 Real Numbers

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Whole Numbers Always begin with 0 { 0, 1, 2, 3, 4, 5,.....} The set of Whole Numbers is the same as Natural except that it includes 0. The way to remember it is think “0” in “whole”

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Integers The set of all natural numbers and their additive inverses (opposites) and 0. {.... -3, -2, -1, 0, 1, 2, 3,....} Does not include fractions or decimals

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Rational Numbers Numbers that can be expressed as the ratio (fraction) of two integers, a/b where b ≠ 0. Decimal representations of rational numbers either terminate or repeat. Examples: – 2.375, can be read as 2 and 375 thousandths and written as 2 375/1000, (terminating decimal) – 4, can be written as 16/4, 4/1, 8/2 – −0.25, can be read as negative 25 one-hundredths and written as - 25/100 – 0.14, repeating decimal and can be written as 14/99

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Irrational Numbers Numbers that cannot be expressed as a ratio (fraction) of two integers. Their decimal representations neither terminate nor repeat. Decimals that go on forever without repeating a pattern. Examples: –– – 3– 3 –0.14114111411114…

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Real Number System Irrational Numbers Rational Numbers Integers - + Whole Numbers “0” Natural Numbers “Counting” Fractions

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Rational Numbers Any number that can be written as a fraction a where be can not equal 0. b Irrational Numbers REAL NUMBER SYSTEM Integers All Positive Numbers and their opposites including 0. Whole Numbers All positive numbers plus 0. Natural Numbers

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Questions Determine if the following statements are true or false and give a short reason why: –Every integer is a rational number. –Every rational number is an irrational number. –Every natural number is an integer. –Every integer is a natural number.

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