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# Classification of the Real Number System

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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1. Classification of the Real Number System

2. 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.

3. 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.

4. 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.

5. 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

6. 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

7. 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

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

9. 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

10. 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

11. 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

12. 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.

13. 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

14. -4.2 -6 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3

15. -4.2 -6 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3

16. -4.2 -6 2.5 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3

17. -4.2 -6 2.5 -6 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3

18. -4.2 -6 2.5 -6 -4.2 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3

19. -4.2 -6 2.5 -6 -4.2 2.5 5 4 3 2 6 1 -1 0 -4 -2 -5 -6 -3

20. 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.

21. Rational Numbers Not Considered Real -5 Integers -5 -5 Whole 18% Natural/Counting 26 Irrational

22. Rational Numbers Not Considered Real -5 Integers -5 -5 Whole 18% Natural/Counting 26 Irrational

23. Rational Numbers Not Considered Real -5 18% Integers -5 -5 Whole 18% Natural/Counting 26 Irrational

24. Rational Numbers Not Considered Real -5 18% Integers -5 -5 Whole 18% Natural/Counting 26 Irrational

25. Rational Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

26. Rational Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

27. Rational Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

28. Rational Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

29. Rational Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

30. Rational Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

31. Rational Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

32. Rational Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

33. Rational Numbers Not Considered Real 26 -5 18% Integers 26 -5 -5 Whole 26 18% Natural/Counting 26 26 Irrational

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