1.1 real number
TRANSCRIPT
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1.1 Learning Outcomes
To define and understand natural, whole, integers, primenumbers, rational and irrational numbers.
To represent rational and irrational numbers in decimalform,
To represent the relationship of number setsandproperties in a real numbers
To understand open, closed and half open intervalrepresentations on the number line.
To simplify union, , and intersection, , of two or moreintervals with the aid of number line.
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the set of counting numbersN= {1, 2, 3 }
Prime numbers : the number greaterthan 1 and can be divided by itselfonly.
Prime number = {2, 3, 5, 7 }
1.1.1 Natural Numbers
1.1.2 Prime Numbers
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The natural numbers, togetherwith the number 0
W= {0, 1, 2, 3 }
Do you know what is the wholenumbers?
1.1.3 Whole Numbers
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1.1.4 Integers
The whole numbers together withthe negative of counting numbersform the set of integers anddenoted by Z.
Z= {, -3, -2, -1, 0, 1, 2, 3 }
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1.1.5 Rational numbers
A rational number is any number thatcan be represented as a ratio(quotient) of two integers and can be
written as
,,; bZbab
aQ
Rational number can be expressedas terminating or repeatingdecimals eg. 0.3333,0.5
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1.1.6 Irrational number
The number cannot be written as aquotient and non repeating decimal
number.
Eg : 0.452138, ,3
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Rational numbers Irrational Numbers
1415.3
414.12
25.04
1
...363636.011
4
...333.03
1
Eg.
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Numbers that either
rational or irrational arecalled real number, R.
1.1.7 Real Numbers
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Exercise:
For the set of {-5, -3, -1, 0, 3, 8},identify the set of
(a) natural numbers :
N ={3,8}
(b) whole numbersW = {0,3,8}
(c) prime numbersprime number={3}
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(d) even numbers
even number = {0,8}
(e) negative integers= {-5, -3,-1}
(f) odd numbersodd number ={-5, -3, -1, 3}
Z
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Express each of this number as a quotient(a) 1.555 (b) 5.45959
(a) Let x= 1.555 (1)
(1) 10 10x= 15.555 (2)thus, (2) (1), 9x= 14
x= 9
14
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Let x= 5.45959 (1)
(1) 10 10x= 54.5959 (2)
(2) 100 100x= 5459.5959 (3)
therefore, (3) (2), 990x= 5405
x =
x =
990
5405
198
1081
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Properties of Real
NumbersAddition Multiplication
1. Closure a+ b= c, cR
6 + 7 = 13 R
ab= d, dR
6 7 = 42 R
2. Commutative a+ b= b+ a
2 + 5 = 5 + 2
ab= ba
2 5 = 5 2
3. Associativea+ b) + c= a+ (b+ c)
(1 + 3) + 2 = 1 + (3 + 2)
(ab)c= a(bc)
(4 3) 2 = 4 (3 2)
4. Distributivea(b+ c) = ab+ ac
4 (2 + 3) = 4 2 + 4 3
5. Identity a+ 0 = 0 + a= a5 + 0 = 0 + 5 = 5
a 1 = 1 a= a3 1 = 1 3 = 3
6. Inversea+ (a) = 0 = (a) + a
7 + (7) = 0 = (7) + 7515
5
1
aa
a
11
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Example
Write down the type of properties for this statement
(a) 3 + 4x =4x +3Commutative
(b) x(y + z) = xy + xzDistributive
(c) 3ab +0 =3ab
Identity for Addition Operation
(d) 2(3n) =(2(3)) nAssociative for Multiply Operation
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ExampleGiven a, bR, ab = 1. Prove that a =
b
1
.
Solution
Given ab =1(ab)
b
1= 1
b
1
a
b
b1
=
b
1 (associative and identity)
b
1 (Inverse)
b
1 (identity)
a 1 =
a=
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1.1.8 Number Line
The set of numbers that corresponds to all point on number lines iscalled the set ofreal number.
The real numbers on the number line are ordered in increasingmagnitude from the left to the right
3
2
4 -3 -2 -1 0 1 2 3 4
3.5
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Type of intervala) Open interval (a, b) or
{x : a < x < b}
b) Closed interval [a, b] or{x: a x b}
b) Half closed interval,(i) (a, b] or {x : a < x b}
(ii) [a, b) or {x : a x < b}
a b
a b
a b
a b
i)
ii)
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ExampleList the number described and graph the numbers on anumber line.a)The whole number less than 4
W = {0 , 1, 2 , 3}
-3 -2 -1 0 1 2 3 4
b) The integer between 3 and 9Z = {4, 5, 6, 7, 8}
2 3 4 5 6 7 8 9
-5 -4 -3 -2 -1 0 1 2
c) The integers greater than -3Z = {-2, -1, 0 }
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Example
Represent the following interval on the real number line andstate the type of the interval.(a) [-1, 4] (b) {x: 2 x 5}(c) [2, ) (d) {x: x 0)
-1 4
a) closed interval
2 5open intervalb)
c)
2
half-open interval
d)0
half-open interval
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Intersection and union operation
Example :
Given A = [1 , 6) and B = (2, 4),
-2 -1 0 1 2 3 4 5 6
Union ()
Intersection ()
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Example:Solve the following using the number line [0, 5) (4, 7)
0 4 5 7
[0, 5) (4, 7) = [0, 7)
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Solve the following using the number line(, 0] [0, )
-
0
(, 0] [0, ) = (, ) = R
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Solve the following using the number line(4, 2) (0, 4] [2, 2)
Consider(4, 2) (0, 4]
-4 0 2 4
(4, 2) (0, 4] = (4, 4]
-4 -2 0 2 4
(4, 4] [2, 2) = [2, 2)
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Example
Given A = {x: -2 x 5} and B = {x: 0 x 7}.
Show that A B = (0, 5].
2 0 5 7
(2, 5] (0, 7] = (0, 5]
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So today, you learn about...
IntersectionOr Union
Number line
Types OfReal Numbers
Real Numbers
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