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Classification of the Real Number System. Not Real. 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. .

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## Classification of the Real Number System

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**Not Real**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**or Terminating decimal Ratio of two Number repeating decimal integers 5.000 terminating .250 terminating 20% .20 terminating = repeating - 8 - 8.0 terminating - 2.5 - 2.50 terminating - 2 = - - 6.0 terminating 0 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 --- 1. 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**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 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**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**Pi 𝞹 3.1415926535897932384626433832795… (and more) 4.47213594… 0.8660254… 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. 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3**-4.2**-6 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3**-4.2**-6 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3**-4.2**-6 2.5 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3**-4.2**-6 2.5 -6 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3**-4.2**-6 2.5 -6 -4.2 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3**-4.2**-6 2.5 -6 -4.2 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3**Numbers Not Considered Real**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 -5 -5 Whole 18% Natural/Counting 26 Irrational**Rational**Numbers Not Considered Real -5 Integers -5 -5 Whole 18% Natural/Counting 26 Irrational**Rational**Numbers Not Considered Real -5 18% Integers -5 -5 Whole 18% Natural/Counting 26 Irrational**Rational**Numbers Not Considered Real -5 18% Integers -5 -5 Whole 18% Natural/Counting 26 Irrational**Rational**Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational**Rational**Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational**Rational**Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational**Rational**Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational**Rational**Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational**Rational**Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational**Rational**Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational**Rational**Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational**Rational**Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

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