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# Classification of the Real Number System

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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.

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.

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.

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

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

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

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

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

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

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

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.

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

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

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

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

− 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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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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