Lecture 01 reals number system
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Transcript of Lecture 01 reals number system
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Engr. Mexieca M. Fidel
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THE REAL NUMBER SYSTEM
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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.
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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”
WRITE SETS USING SET NOTATION
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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}
WRITE SETS USING SET NOTATION
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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,…}
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1-1 Using a number line
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, …}
-2 -1 0 1 2 3 4 5
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1-1 Using a number line
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.
-2 -1 0 1 2 3 4 5
coordinate
Graph of -1
o12
114
o o
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1-1 Using a number line
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
-2 -1 0 1 2 3 4 5
coordinate
Graph of -1
o12
114
o o2
7
4 16 o o oo o
Circumferencediameter
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OPERATIONS INVOLVING SETS
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REAL NUMBERS (R)Definition:
REAL NUMBERS (R)- Set of all rational and irrational numbers.
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SUBSETS of RDefinition:
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}
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SUBSETS of RDefinition:
INTEGERS (Z)- numbers that consist of positive integers, negative integers, and zero,- {…, -2, -1, 0, 1, 2 ,…}
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SUBSETS of RDefinition:
NATURAL NUMBERS (N)- counting numbers- positive integers- {1, 2, 3, 4, ….}
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SUBSETS of RDefinition:
WHOLE NUMBERS (W)- nonnegative integers- { 0 } {1, 2, 3, 4, ….}- {0, 1, 2, 3, 4, …}
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SUBSETS of RDefinition:
IRRATIONAL NUMBERS (Q´)- non-terminating and non-repeating decimals- transcendental numbers- Ex: {pi, sqrt 2, -1.436512…..}
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The Set of Real Numbers
Q
Q‘(Irrational Numbers)Q(Rational Numbers)
Z(Integers)
W(whole numbers)
N(Natural numbers)
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PROPERTIES of RDefinition:
CLOSURE PROPERTYGiven real numbers a and b,
Then, a + b is a real number (+), or a x b is a real number (x).
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PROPERTIES of RExample 1:
12 + 3 is a real number. Therefore, the set of reals is CLOSED with respect to addition.
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PROPERTIES of RExample 2:
12 x 4.2 is a real number. Therefore, the set of reals is CLOSED with respect to multiplication.
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PROPERTIES of RDefinition:
COMMUTATIVE PROPERTYGiven real numbers a and b,Addition: a + b = b + aMultiplication: ab = ba
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PROPERTIES of RExample 3:Addition:
2.3 + 1.2 = 1.2 + 2.3Multiplication:
(2)(3.5) = (3.5)(2)
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PROPERTIES of RDefinition:
ASSOCIATIVE PROPERTYGiven real numbers a, b and c,
Addition: (a + b) + c = a + (b + c)Multiplication: (ab)c = a(bc)
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PROPERTIES of RExample 4:Addition:
(6 + 0.5) + ¼ = 6 + (0.5 + ¼) Multiplication:
(9 x 3) x 4 = 9 x (3 x 4)
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PROPERTIES of RDefinition:
DISTRIBUTIVE PROPERTY of MULTIPLICATION OVER ADDITIONGiven real numbers a, b and c,
a (b + c) = ab + ac
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PROPERTIES of RExample 5:4.3 (0.11 + 3.02) = (4.3)(0.11) + (4.3)(3.02)
Example 6:2x (3x – b) = (2x)(3x) + (2x)(-b)
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PROPERTIES of RDefinition:
IDENTITY PROPERTYGiven a real number a,Addition: 0 + a = a Multiplication: 1 x a = a
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PROPERTIES of RExample 7:Addition:
0 + (-1.342) = -1.342 Multiplication:
(1)(0.1234) = 0.1234
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PROPERTIES of RDefinition:
INVERSE PROPERTYGiven a real number a,Addition: a + (-a) = 0 Multiplication: a x (1/a) = 1
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PROPERTIES of RExample 8:Addition:
1.342 + (-1.342) = 0 Multiplication:
(0.1234)(1/0.1234) = 1
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EXERCISESTell 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)]107. [(4)(10)]10 = 4[(10)(10)]8. 3 + 0.33 is a real number
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TRUE OR FALSE 1. The set of WHOLE numbers is closed with respect to multiplication.
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TRUE OR FALSE2. The set of NATURAL numbers is closed with respect to multiplication.
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TRUE OR FALSE3. The product of any two REAL numbers is a REAL number.
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TRUE OR FALSE4. The quotient of any two REAL numbers is a REAL number.
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TRUE OR FALSE5. Except for 0, the set of RATIONAL numbers is closed under division.
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TRUE OR FALSE6. Except for 0, the set of RATIONAL numbers contains
the multiplicative inverse for each of its members.
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TRUE OR FALSE7. The set of RATIONAL numbers is associative under multiplication.
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TRUE OR FALSE8. The set of RATIONAL numbers contains the additive inverse for each of its members.
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TRUE OR FALSE9. The set of INTEGERS is commutative under subtraction.
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TRUE OR FALSE10. The set of INTEGERS is closed with respect to division.