![EXERCISES
Tell which of the properties of real
numbers justifies each of the following
statements.
1. (2)(3) + (2)(5) = 2 (3 + 5)
2. (10 + 5) + 3 = 10 + (5 + 3)
3. (2)(10) + (3)(10) = (2 + 3)(10)
4. (10)(4)(10) = (4)(10)(10)
5. 10 + (4 + 10) = 10 + (10 + 4)
6. 10[(4)(10)] = [(4)(10)]10
7. [(4)(10)]10 = 4[(10)(10)]
8. 3 + 0.33 is a real number](https://image.slidesharecdn.com/lecture01realsnumbersystem-141110052603-conversion-gate01/85/Lecture-01-reals-number-system-40-320.jpg)
Lecture 01 reals number system
- 1. Engr. Mexieca M. Fidel
- 2. THE REAL NUMBER SYSTEM
- 3. WRITE SETS USING SET NOTATION A set is a collection of objects called the elements or members of the set. Set braces { } are usually used to enclose the elements. In Algebra, the elements of a set are usually numbers. • Example 1: 3 is an element of the set {1,2,3} Note: This is referred to as a Finite Set since we can count the elements of the set. • Example 2: N= {1,2,3,4,…} is referred to as a Natural Numbers or Counting Numbers Set. • Example 3: W= {0,1,2,3,4,…} is referred to as a Whole Number Set.
- 4. WRITE SETS USING SET NOTATION A set is a collection of objects called the elements or members of the set. Set braces { } are usually used to enclose the elements. • Example 4: A set containing no numbers is shown as { } Note: This is referred to as the Null Set or Empty Set. Caution: Do not write the {0} set as the null set. This set contains one element, the number 0. • Example 5: To show that 3 “is a element of” the set {1,2,3}, use the notation: 3 {1,2,3}. Note: This is also true: 3 N • Example 6: 0 N where is read as “is not an element of”
- 5. WRITE SETS USING SET NOTATION Two sets are equal if they contain exactly the same elements. (Order doesn’t matter) • Example 1: {1,12} = {12,1} • Example 2: {0,1,3} {0,2,3}
- 6. In Algebra, letters called variables are often used to represent numbers or to define sets of numbers. (x or y). The notation {x|x has property P}is an example of “Set Builder Notation” and is read as: {x x has property P} the set of all elements x such that x has a property P • Example 1: {x|x is a whole number less than 6} Solution: {0,1,2,3,4,5} • Example 2: {x|x is a natural number greater than 12} Solution: {13,14,15,…}
- 7. 1-1 Using a number line -2 -1 0 1 2 3 4 5 One way to visualize a set a numbers is to use a “Number Line”. • Example 1: The set of numbers shown above includes positive numbers, negative numbers and 0. This set is part of the set of “Integers” and is written: I = {…, -2, -1, 0, 1, 2, …}
- 8. 1-1 Using a number line Graph of -1 o 1 2 11 4 o o -2 -1 0 1 2 3 4 5 coordinate Each number on a number line is called the coordinate of the point that it labels, while the point is the graph of the number. • Example 1: The fractions shown above are examples of rational numbers. A rational number is one than can be expressed as the quotient of two integers, with the denominator not 0.
- 9. 1-1 Using a number line Graph of -1 4 16 2 o o o o o -2 -1 0 1 2 3 4 5 coordinate Decimal numbers that neither terminate nor repeat are called “irrational numbers”. • Example 1: Many square roots are irrational numbers, however some square roots are rational. • Irrational: Rational: 2 7 4 16 o 1 2 11 4 o o 7 Circumference diameter
- 20. REAL NUMBERS (R) Definition: REAL NUMBERS (R) - Set of all rational and irrational numbers.
- 21. SUBSETS of R Definition: RATIONAL NUMBERS (Q) - numbers that can be expressed as a quotient a/b, where a and b are integers. - terminating or repeating decimals - Ex: {1/2, 55/230, -205/39}
- 22. SUBSETS of R Definition: INTEGERS (Z) - numbers that consist of positive integers, negative integers, and zero, - {…, -2, -1, 0, 1, 2 ,…}
- 23. SUBSETS of R Definition: NATURAL NUMBERS (N) - counting numbers - positive integers - {1, 2, 3, 4, ….}
- 24. SUBSETS of R Definition: WHOLE NUMBERS (W) - nonnegative integers - { 0 } {1, 2, 3, 4, ….} - {0, 1, 2, 3, 4, …}
- 25. SUBSETS of R Definition: IRRATIONAL NUMBERS (Q´) - non-terminating and non-repeating decimals - transcendental numbers - Ex: {pi, sqrt 2, -1.436512…..}
- 26. The Set of Real Numbers Q Q‘ (Irrational Numbers) Q (Rational Numbers) Z (Integers) W (whole numbers) N (Natural numbers)
- 27. PROPERTIES of R Definition: CLOSURE PROPERTY Given real numbers a and b, Then, a + b is a real number (+), or a x b is a real number (x).
- 28. PROPERTIES of R Example 1: 12 + 3 is a real number. Therefore, the set of reals is CLOSED with respect to addition.
- 29. PROPERTIES of R Example 2: 12 x 4.2 is a real number. Therefore, the set of reals is CLOSED with respect to multiplication.
- 30. PROPERTIES of R Definition: COMMUTATIVE PROPERTY Given real numbers a and b, Addition: a + b = b + a Multiplication: ab = ba
- 31. PROPERTIES of R Example 3: Addition: 2.3 + 1.2 = 1.2 + 2.3 Multiplication: (2)(3.5) = (3.5)(2)
- 32. PROPERTIES of R Definition: ASSOCIATIVE PROPERTY Given real numbers a, b and c, Addition: (a + b) + c = a + (b + c) Multiplication: (ab)c = a(bc)
- 33. PROPERTIES of R Example 4: Addition: (6 + 0.5) + ¼ = 6 + (0.5 + ¼) Multiplication: (9 x 3) x 4 = 9 x (3 x 4)
- 34. PROPERTIES of R Definition: DISTRIBUTIVE PROPERTY of MULTIPLICATION OVER ADDITION Given real numbers a, b and c, a (b + c) = ab + ac
- 35. PROPERTIES of R Example 5: 4.3 (0.11 + 3.02) = (4.3)(0.11) + (4.3)(3.02) Example 6: 2x (3x – b) = (2x)(3x) + (2x)(-b)
- 36. PROPERTIES of R Definition: IDENTITY PROPERTY Given a real number a, Addition: 0 + a = a Multiplication: 1 x a = a
- 37. PROPERTIES of R Example 7: Addition: 0 + (-1.342) = -1.342 Multiplication: (1)(0.1234) = 0.1234
- 38. PROPERTIES of R Definition: INVERSE PROPERTY Given a real number a, Addition: a + (-a) = 0 Multiplication: a x (1/a) = 1
- 39. PROPERTIES of R Example 8: Addition: 1.342 + (-1.342) = 0 Multiplication: (0.1234)(1/0.1234) = 1
- 40. EXERCISES Tell which of the properties of real numbers justifies each of the following statements. 1. (2)(3) + (2)(5) = 2 (3 + 5) 2. (10 + 5) + 3 = 10 + (5 + 3) 3. (2)(10) + (3)(10) = (2 + 3)(10) 4. (10)(4)(10) = (4)(10)(10) 5. 10 + (4 + 10) = 10 + (10 + 4) 6. 10[(4)(10)] = [(4)(10)]10 7. [(4)(10)]10 = 4[(10)(10)] 8. 3 + 0.33 is a real number
- 41. TRUE OR FALSE 1. The set of WHOLE numbers is closed with respect to multiplication.
- 42. TRUE OR FALSE 2. The set of NATURAL numbers is closed with respect to multiplication.
- 43. TRUE OR FALSE 3. The product of any two REAL numbers is a REAL number.
- 44. TRUE OR FALSE 4. The quotient of any two REAL numbers is a REAL number.
- 45. TRUE OR FALSE 5. Except for 0, the set of RATIONAL numbers is closed under division.
- 46. TRUE OR FALSE 6. Except for 0, the set of RATIONAL numbers contains the multiplicative inverse for each of its members.
- 47. TRUE OR FALSE 7. The set of RATIONAL numbers is associative under multiplication.
- 48. TRUE OR FALSE 8. The set of RATIONAL numbers contains the additive inverse for each of its members.
- 49. TRUE OR FALSE 9. The set of INTEGERS is commutative under subtraction.
- 50. TRUE OR FALSE 10. The set of INTEGERS is closed with respect to division.