number system -real number (14.6.2012)
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NUMBER SYSTEM
Real number
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Learning Outcomes
(a) To define and understand natural, whole, integers, primenumbers, rational and irrational numbers.(b) To represent rational and irrational numbers in decimal
form,(c) To represent the relationship of number sets in a real
numbers
(d) To state clearly the properties of real numbers(e) To understand open, closed and half open interval and their
representations on the number line.(f) To understand that the end point of and open interval on
the number line are usually represented as empty circles, whereas the end points of a closed interval are
represented by dense circles, .(g) To simplify union, , and intersection, , of two or more
intervals with the aid of number line.
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the set of counting numbersN= {1, 2, 3 }
Prime numbers : the numbergreater than 1 and can bedivided by itself only.
Prime number = {2, 3, 5, 7 }
Natural Numbers
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The natural numbers,together with the number 0
W= {0, 1, 2, 3 }.
Do you know what is the
whole numbers?
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The Integers
The whole numbers together with the
negative of counting numbers form the setof integers anddenoted by Z.
Z= {, -3, -2, -1, 0, 1, 2, 3 }
Integers
Positive integersZ+= {1, 2, 3 }
0Negative integersZ-= {, -3, -2, -1}.
Integers
Odd numbers{2k+ 1, kZ}
Even numbers{2k, kZ}
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The rational numbers
A rational numberis any numberthat can be represented as aratio (quotient) of two integers
and can be written as
0,,; bZbab
aQ
Rational number can be expressedas terminating or repeating
decimals
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Irrational number
The number cannot be writtenas a quotient and non repeating
decimal number.
Eg : 0.452138, ,3
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Rationalnumbers
IrrationalNumbers
14159265.3
414214.12
25.04
1
...363636.011
4
...333.03
1
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Numbers that eitherrational or irrational arecalled real number, R.
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Q
R
Q
Z
W
N
Relationship of Number Sets
From the diagram, we can see that : 1. N W Z Q R2. Q = RQ
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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 aquotient
(a) 1.555 (b) 5.45959
(a) Let x= 1.555 (1)
(1) 10 10x= 15.555 (2)thus, (2) (1), 9x= 14
x=
914
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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 =
=
990
5405
198
1081
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Properties ofReal Numbers
Addition Multiplication
1. Closure a+ b= c, cR
6 + 7 = 13 R
ab= d, dR
6 7 = 42 R
2. Commutative a+ b= b+ a2 + 5 = 5 + 2
ab= ba2 5 = 5 2
3. Associative a+ b) + c= a+ (b+ c)(1 + 3) + 2 = 1 + (3 + 2)
(ab)c= a(bc)(4 3) 2 = 4 (3 2)
4. Distributive a(b+ c) = ab+ ac4 (2 + 3) = 4 2 + 4 3
5. Identity a+ 0 = 0 + a= a5 + 0 = 0 + 5 = 5
a1 = 1 a= a3 1 = 1 3 = 3
6. Inverse a+ (a) = 0 = (a) + a7 + (7) = 0 = (7) + 7
5
1515
5
1
aaa
a
11
1
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ExampleWrite down the type of properties for this statement
(a) 3 + 4x = 4x + 3
Commutative
(b)x (y + z)= xy + xz
Distributive
(c) 3ab + 0= 3ab
Identity for Addition Operation
(d) 2(3n)= (2(3))n
Associative for Multiply Operation
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Example
Given a, bR, ab= 1. Prove that a =
b
1
.
Solution
Given ab = 1(ab)
b
1= 1
b
1
a
b
b 1
=
b
1 (associative and identity)
b
1 (Inverse)
b
1 (identity)
a1 =
a=
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The number line
The set of numbers that corresponds to all point on numberlines is called the set ofreal number.
The real numbers on the number line are ordered inincreasing magnitude 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 4W = {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
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ExampleList the number described and graph the numbers on anumber line.a) The integer between 3 and 9 :
Z = {4, 5, 6, 7, 8}
2 3 4 5 6 7 8 9
b) The integers greater than -3Z = {-2, -1, 0 }
-5 -4 -3 -2 -1 0 1 2
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ExampleRepresent the following interval on the real number line andstate the type of the interval.(a) [-1, 4] (b) {x: 2 x5}(c) [2, ) (d) {x: x 0,xR}
-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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ExampleSolve 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(b) (, 5) (1, 9)
1 5 9
(, 5) (1, 9) = (1, 5)
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Solve the following using the number line(c) (, 0] [0, )
-
0
(, 0] [0, ) = (, ) = R
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Example
Given A = {x: -2 x5} and B = {x: 0 x7}.Show that A B = (0, 5].
2 0 5 7 (2, 5] (0, 7] = (0, 5]