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**Classification of the Real Number System**

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Real Not Real Rational Irrational Rational - any number that can be written as the ratio of two integers, which consequently can be expressed as a terminating or repeating decimal. Irrational - numbers that cannot be written as a ratio of two integers.

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Real Rational Irrational Integers Integers are positive and negative whole numbers and zero such as … -4, -3, -2, -1, 0, 1, 2, 3, 4 and so on. Important Tip Integers do not have any fractional parts. So numbers such a ½, .3, 2 ¼ , 25% etc are not integers because they involve fractional parts.

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Also … When determining if a number is rational the number must be able to be written in such a way that the numerator and denominator is a positive or negative whole number. Additionally … The numerator can be zero but not the denominator.

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**Examples of Rational Numbers**

Terminating decimal or Number Ratio of two integers repeating decimal 5.000 terminating .250 terminating 20% .20 terminating = repeating - 8 - 8.0 terminating - 2.5 - 2.50 terminating - 2 = - - 6.0 terminating 0.0 terminating

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**The set of rational numbers has subsets**

Some common subsets of rational numbers are Natural/counting numbers Whole numbers Integers Some numbers fall into more than one category

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Real Natural/counting numbers (N) are positive whole numbers beginning with 1. A way to remember natural / counting numbers is to think about what number you begin counting with So natural / counting numbers are numbers such as 1, 2, 3, 4, etc. Natural

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Real Whole numbers (W) include ALL counting numbers and 0. So whole numbers are 0, 1, 2, 3, 4, etc. Whole Natural

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**Real Integers Whole Natural**

Integers (Z) were explained previously but to recall they include all natural/counting numbers and whole numbers. They are positive and negative whole numbers and 0 such as … -4, -3, -2, -1, 0, 1, 2, 3, 4 … Real Integers Whole Natural

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**Real Rational Integers Whole Natural**

Rational Numbers (Q) recall that they are zero and all positive and negative numbers that can be expressed as a ratio of two integers (with no zero in the denominator), including integers, whole numbers, and natural/counting numbers. Rational Integers Whole Natural

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**Real Rational Irrational Integers Whole Natural**

Irrational Numbers (I) recall that they are real numbers that are not rational and cannot be written as a ratio of integers. Rational Irrational Integers Whole Natural

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**Examples of Irrational Numbers**

3 2 20 Pi 𝞹 … (and more) … … Irrational numbers are considered real numbers. The real number system can be divided into two categories – rational and irrational. Many students tend to think that irrational numbers are not real. This is not true. Irrational numbers ARE real but just are expressed differently than rational numbers.

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Basically in order to determine if a number is real, ask yourself if the numbers can be placed on a number line. If the number can be placed on a number line or be ordered, then the number is real. -1 -2 -3 -4 -5 -6 1 2 3 4 5 6

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𝟑𝟓 2.5 -6 -4.2 𝟏𝟔 − 𝟑 − 𝟑 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6

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𝟑𝟓 2.5 -6 -4.2 𝟏𝟔 − 𝟑 𝟑𝟓 − 𝟑 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6

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− 𝟑 𝟑𝟓 2.5 -6 -4.2 𝟏𝟔 𝟑𝟓 − 𝟑 2.5 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6

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− 𝟑 𝟑𝟓 2.5 -6 -4.2 𝟏𝟔 -6 𝟑𝟓 − 𝟑 2.5 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6

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𝟑𝟓 2.5 -6 -4.2 𝟏𝟔 − 𝟑 -6 𝟑𝟓 -4.2 − 𝟑 2.5 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6

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𝟑𝟓 2.5 -6 -4.2 𝟏𝟔 − 𝟑 -6 -4.2 − 𝟑 2.5 𝟏𝟔 𝟑𝟓 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6

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**𝒙 𝟐 = −𝟏 𝒙 𝟐 = −𝟏 𝒙 = −𝟏 Numbers Not Considered Real 𝟏𝟖 𝟎 𝟑 𝟎 −𝟕.𝟑 𝟎**

𝒙 𝟐 = −𝟏 𝟏𝟖 𝟎 𝟑 𝟎 −𝟕.𝟑 𝟎 𝒙 𝟐 = −𝟏 𝒙 = −𝟏 These numbers are undefined because zero is in the denominator and cannot be considered a real number. They are not numbers at all. The square root of any negative number are numbers not considered real.

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Rational Numbers Not Considered Real -5 Integers −25 -5 -5 7 0 − 2 3 Whole 18% − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 26 Irrational 0 20 121 3

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Rational Numbers Not Considered Real -5 𝟕 𝟎 Integers −25 -5 -5 7 0 − 2 3 Whole 18% − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 26 Irrational 0 20 121 3

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Rational Numbers Not Considered Real -5 18% 7 0 Integers −25 -5 -5 7 0 − 2 3 Whole 18% − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 26 Irrational 0 20 121 3

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Rational 8 2 Numbers Not Considered Real -5 18% 7 0 Integers −25 -5 8 2 -5 7 0 − 2 3 Whole 18% 8 2 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 Irrational 0 20 121 3

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Rational 8 2 26 Numbers Not Considered Real -5 18% 7 0 Integers 26 −25 -5 8 2 -5 7 0 − 2 3 Whole 26 18% 8 2 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 26 Irrational 0 20 121 3

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Rational 8 2 121 3 26 Numbers Not Considered Real -5 18% 7 0 Integers 26 −25 -5 8 2 -5 7 0 − 2 3 Whole 26 18% 8 2 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 26 Irrational 0 20 121 3

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Rational 8 2 121 3 26 Numbers Not Considered Real -5 18% 7 0 −25 Integers 26 −25 -5 8 2 -5 7 0 − 2 3 Whole 26 18% 8 2 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 26 Irrational 0 20 121 3

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Rational 8 2 121 3 26 Numbers Not Considered Real -5 18% 7 0 −25 Integers 26 −25 -5 8 2 -5 7 0 − 2 3 Whole 26 18% 8 2 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 26 Irrational 0 20 121 3 − 67

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Rational 8 2 121 3 − 𝟐 𝟑 26 Numbers Not Considered Real -5 18% 7 0 −25 Integers 26 −25 -5 8 2 -5 7 0 − 2 3 Whole 26 18% 8 2 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 26 Irrational 0 20 121 3 − 67

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Rational 8 2 121 3 − 2 3 26 Numbers Not Considered Real -5 18% 7 0 −25 −2. 73 Integers 26 −25 -5 8 2 -5 7 0 − 2 3 Whole 26 18% 8 2 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 26 Irrational 0 20 121 3 − 67

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Rational 8 2 121 3 − 2 3 26 Numbers Not Considered Real -5 18% 7 0 −25 −2. 73 Integers 26 −25 -5 8 2 -5 7 0 − 2 3 Whole 26 18% 8 2 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 26 Irrational 0 20 𝟐𝟎 𝟐 121 3 − 67

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Rational 8 2 121 3 − 2 3 𝟏 𝟑 26 Numbers Not Considered Real -5 18% 7 0 −25 −2. 73 Integers 26 −25 -5 8 2 -5 7 0 − 2 3 Whole 26 18% 8 2 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 26 Irrational 0 20 20 2 121 3 − 67

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Rational 8 2 121 3 − 2 3 1 3 26 Numbers Not Considered Real -5 18% 7 0 −25 𝟎 𝟐𝟎 −2. 73 Integers 26 −25 𝟎 𝟐𝟎 -5 8 2 -5 7 0 − 2 3 Whole 26 18% 8 2 𝟎 𝟐𝟎 − 67 −2. 73 8 2 Natural/Counting 20 2 1 3 8 2 26 26 Irrational 0 20 20 2 121 3 − 67

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