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Real numbers
This document defines and provides examples of different types of real numbers including rational and irrational numbers. It discusses how rational numbers can be represented as fractions or decimals, and how to convert between fraction and decimal representations. Irrational numbers are defined as having non-terminating, non-repeating decimal representations. The key types of real numbers - natural numbers, integers, rational numbers, and irrational numbers - are related in a Venn diagram with their union being the set of all real numbers.
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Real numbers
- 1. Real Numbers www.slideshare.net/reycastro1 The Natureof Modern Mathematics
- 2. Introduction: RATIONALS The society thatwe have been considering has used numbers only as they relate to practical problems. However, this need not be the case, since numbers can be appreciated for their beauty and interrelationships.
- 3. DEFINITION: The rational number 𝟒 𝟓 isinterpreted as the quotient obtained when 4 is divided by 5. RATIONAL NUMBERS The rational numbers, denoted by ℚ, are the set of all numbers of the form 𝓹 𝓺 , where 𝓹 and 𝓺 are integers and 𝓺 ≠ 𝟎.
- 4. Thus, 𝟓 𝟗 , −𝟏𝟏 𝟏𝟑 , 𝟓 𝟏 , 𝟎 𝟏𝟕 , −𝟏𝟎 𝟓 , and 𝟐𝟓 𝟓 are allelements of ℚ. RATIONAL NUMBERS
- 5. Notice that, 𝟓 𝟏 , 𝟐𝟓 𝟓 , −𝟏𝟎 −𝟐 , … representthe same numbers. 𝒂 𝒃 = 𝒄 𝒅 means ad = bc. RATIONAL NUMBERS
- 6. Thus we seethat, 𝟐𝟓 𝟓 = −𝟏𝟎 −𝟐 , since 25(-2) = 5(-10) -50 = -50 Also, 125/70 ≠ 73/41 why? RATIONAL NUMBERS
- 7. If a/b andc/d are rational numbers, then 𝒂 𝒃 + 𝒄 𝒅 = 𝒂𝒅 + 𝒃𝒄 𝒃𝒅 . where a, b, c, d are integers, b and d nonzero. ADDITION OF RATIONALS
- 8.
- 9. If a/b andc/d are rational numbers, then 𝒂 𝒃 − 𝒄 𝒅 = 𝒂𝒅 − 𝒃𝒄 𝒃𝒅 . where a, b, c, d are integers, b and d nonzero. SUBTRACTION OF RATIONALS Example: 1. 𝟐 𝟑 − −𝟏 𝟐 = 2. 𝟒 𝟗 − 𝟕 𝟗 =
- 10. If a/b andc/d are rational numbers, then 𝒂 𝒃 ∙ 𝒄 𝒅 = 𝒂𝒄 𝒃𝒅 . MULTIPLICATION OF RATIONALS Example: 1. 𝟒 𝟓 ∙ 𝟕 𝟗 = 2. −𝟑 𝟓 ∙ −𝟏 𝟐 =
- 11. If a/b andc/d are rational numbers, c≠ 0, then 𝒂 𝒃 𝒄 𝒅 = 𝒂𝒅 𝒃𝒄 . DIVISION OF RATIONALS Example: 1. −𝟑 𝟒 ÷ −𝟒 𝟕 = 2. 𝟒 𝟓 ÷ 𝟕 𝟗 =
- 12. A. 1. Findthe sum. a. 𝟐 𝟑 + 𝟕 𝟗 b. −𝟓 𝟕 + 𝟒 𝟑 c. −𝟏𝟐 𝟑𝟓 - 𝟖 𝟏𝟓 d. 𝟒 𝟓 + 𝟐 + −𝟕 𝟐𝟓 e. 2-1 + 2 PROBLEM SET:
- 13. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 𝟐 Review:
- 14. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 𝟐 Review:
- 15. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 𝟐 Review:
- 16. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 𝟐 Review:
- 17. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 𝟐 Review:
- 18. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 0.75 𝟐 Review:
- 19. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 0.75 𝟐 Review:
- 20. Pythagoreans discovered thefamous property of square numbers that today bears Pythagoras’ name. “In any right triangle, the area of the larger square was equal to the sum of the areas of the smaller squares. a2 + b2 = c2 IRRATIONALS
- 21. Example: a). b) 5,12, 13 c) 8, 15, 17 d) Suppose each leg of a right triangle is 1. 3 4 5
- 22. Example: a). b) 5,12, 13 c) 8, 15, 17 d) Suppose each leg of a right triangle is 1. Solution: 12 + 12 = c2 2 = c2 3 4 5 2 = c
- 23. Recall: › An integeris even if it is a multiple of 2. expressed as 2k, for any integer k. › An integer that is not even is said to be odd. expressed as 2k + 1, for any integer k.
- 24. Argument: Is 2 ais a rational a/b, where a and b are integers and b is not 0?
- 25. The necessity ofstill another set of numbers › The set of Irrational Numbers; notation ( Q’ ). 2 is irrational also be 3, 5 and 7 and the square root of any number that is not a perfect square. and so on
- 26. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 0.75 𝟐 Review:
- 27. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 0.75 𝟐 Review:
- 28. Real Numbers Definition: Theset of real numbers, denoted by R, is defined as the union of the set of rationals and the set of irrationals.
- 29. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 0.75 𝟐 Review:
- 30. Given N WZ Q Q’ R 3 0 -3 1/4 -1/3 0.75 𝟐 Review:
- 31. Real numbers thatare rational › Real numbers that are rational can be terminating decimal example: 1 4 = 0.25 and 5 8 = 0.625 Can be represented by repeating decimals example: 2 3 = 0.66 … and 5 11 = 0.4545 …
- 32. Real numbers thatare irrational › Real numbers that are irrational have decimal representations are nonterminating and nonrepeating. example: 2 = 1.414214 … and 𝜋 = 3.1415926 …
- 33. Real Numbers › Wecan say that the real numbers may be classified in several ways. › Any real number is; 1. positive, negative, or zero. 2. a rational number or an irrational number. 3. expressible as terminating, a repeating, or a nonterminating and non-repeating decimal:
- 34. A set ofReal Numbers RQ’Q ZW N -Z0
- 35. Changing From onerepresentation to another › From fraction to decimal Given a fraction a/b, we change it to a decimal; a/b means a divided by b. Example: 1. Change 4/5 to decimal ans. 0.8 2. Change 1/7 to a decimal ans. 0.142857… 3. Change 6 2 9 to a decimal ans. 6. 2
- 36. Changing From onerepresentation to another From decimal to fraction A. If decimal is terminating a) 0.123 means 0.123/1000 b) 6.28 means c) 0.349 means
- 37. Changing From onerepresentation to another From decimal to fraction B. If decimal is repeating procedure: 1. Let n=repeating decimal 2. Multiply both sides of the equation by 10k where k is the number of digits repeating. 3. Subtract the original equation. 4. Solve the resulting equation for n.
- 38. If decimal isrepeating › Example: a) Change 0. 7 to fraction b) Change 0. 123 to fraction c) Change 7. 564 to fraction
- 39. Try it! Problemset › 1. What is the distinguishing characteristic between the rational and irrational numbers? › 2. Draw a Venn Diagram showing the natural numbers, integers, rationals, and irrationals where the universe is the set of reals. › 3.
- 40.