MAT 037 - Chapter 1 - Number and Numbering Systems
Transcript of MAT 037 - Chapter 1 - Number and Numbering Systems
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Chapter 1
Number and Numbering System
1.1 Classification of Real Numbers
The real number is any number that has a decimal representation. Figure 1 illustrates how
the set of numbers are related each others.
Figure 1 : Real numbers and important subsets.
REAL NUMBERS
(R)
Rational Numbers
(Q)
Irrational Numbers
(I)
Integers
(Z)
Negative Integers
(Z-)
Zero Positive Integers
(Z+)
Natural Numbers
(N)
Whole Numbers
(W)
Even Numbers
Odd Numbers
Prime Numbers
Non-Integers
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Real Numbers (R)
All numbers. Rational numbers and Irrational numbers are Real numbers.Rational Numbers (Q)
Numbers that can be represented as a fraction ab
, where a and b are integers and b 0 .
Example 1
Rational numbers,Q ={ }2 15 3 22 5, , 3 , , 25 5 ,... .7 16 1 7 1= = =
Decimal that terminate/end.
Example 2
Rational numbers,Q ={ }4.74, 3.29, 7.895623,... .
Decimal that has repetition of digits.
Example 3
Rational numbers,Q ={ }3.56565656..., 0.0987987987,... .
Irrational Numbers (I)
Numbers that can be represented as non-repeating and non-terminating decimal numbers.
Example 4
Irrational numbers,I ={ }6, , e,1.25648379...,... .
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Example 5
Given the set of numbers { }2, 4, 11, .3 Classify according toa) Real numbers (R)b) Rational numbers (Q)c) Irrational numbers (I)
Solution
Example 6
Given the set of numbers
{ }
22, 5, 9,2
7 , list out the following from the set given.
a) Real numbers (R)b) Rational numbers (Q)c) Irrational numbers (I)
Solution
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Example 7
Find the numbers in the set { }9 , 16, 0, 1.6, 5, 77 + that belong to the specified set.a) Real numbers (R)b) Rational numbers (Q)c) Irrational numbers (I)
Solution
Integers (Z)
Integers are numbers that are classified into negative integers(Z-), zero and positiveintegers(Z+).
Example 8
Negative Integers,Z-={ }..., 5, 4, 3, 2, 1 .
Example 9
Zero={ }0 .
Example 10
Positive Integers, Z+={ }361, 2, 3, 4, 5, , ... .6
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Non Integers
Non-integer means numbers that are not "integers".
Example 11
Non Integers={ }2 5 22, , ,... .3 7 7
Natural Numbers (N)
Counting numbers. All positive integers
Example 12
Natural numbers,N={ 251,2, 3, 4, , 6, 7, ... .5
Whole Numbers(W)
Numbers that start with zero.
Example 13
Whole numbers,W={ 160,1, 2, 3, , 5, ... .4
Even Numbers
Non-zero whole numbers which are divisible by 2 without any remainder. General form : 2n where n=1,2,3,4,...Example 14
Even numbers={ }162, 4, 6, ,... .2
Odd Numbers
Non-zero whole numbers which are not divisible by 2. General form : 2n+1 where n=0,1,2,3,...
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Example 15
Odd numbers={ }181, 3, 5, 7, , ...2
Prime Numbers
Whole numbers that are only divisible by itself and 1 without any remainder.Example 16
Prime numbers={ }2, 3, 5, 7,11,13,17,19,23,29,...
Example 17
Find the numbers in the set { 5-50,0, 3, , ,1.3336 2
that belong to the specified set.
a) Whole numbers (W)b) Natural numbers (N)c) Integers (Z)d) Irrational numbers (Q)e) Real numbers (R)
Solution
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Example 18
Find the numbers in the set{ }35, 0, 4, , 1.8, 2 that belong to the specified set.
a) Whole numbers (W)b) Natural numbers (N)c) Integers (Z)d) Irrational numbers (Q)
Solution
Example 19
Find the numbers in the set
{
1517, , 81, 7, 0
5 + that belong to the specified set.
a) Whole numbers (W)b) Natural numbers (N)c) Integers (Z)d) Prime numbersSolution
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TUTORIAL 1
1. List two elements of each of the following sets of numbersa) Whole numbersb) Natural numbersc) Rational numbersd) Irrational numberse) Prime numbers
2. Determine whether the following statements are TRUE or FALSE.
a) 72
is a rational number.
b) 10 is a non-negative integer.c) Negative four when divided by zero is zero.d) All prime numbers are integers.e) 0 is an integer number.f) An odd number when divided by zero will result in zero.g) 2 is a real number.h) 3 is a rational number.i) Every rational number is an integer.
j) Every irrational number is a real number.k) All integers are natural numbers.l) All odd numbers are whole numbers.m) 1
5is a real number.
n) All integers are whole numberso) 4 is a rational number.p) All prime numbers are odd numbers.q) A positive number when multiplied by a negative number will result in positive
number.
r) All real numbers are rational numbers.s) 2 is a natural number.t) is a Irrational number.u) 1
6is a rational number.
v) 2 + is an irrational number.
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3. Tick the following boxes according to the given numbers.
Number 6 1120
5 0
Prime NumberInteger
Irrational
Number
Whole Number
4. Tick the following boxes according to the given numbers.
Number 7 20 23 2
Natural Number
Non-Integer
Number
Even Number
Prime Number
5. Tick the following boxes according to the given numbers.
Natural
Number
Integer Rational
Number
Irrational
Number
Real
Number
2
5
7
5
25
7
50
423
5
5 22
7
7.5
49
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1.2 Representation of R
Number Line
The number line is an excel
connections across number s
number line.
Through the use of number l
numbers, Integers, Rational n
A number line is a line on wh
Positive numbers are repres
represented on the left of the
Interval Notation
Interval notation translates t
show the interval notation sy
Table 1: Symbol
Symbol
(
)
[
]
Number and
10
al Numbers
lent tool for developing numerical underst
ystems. Real numbers are represented gr
ines you can visually represent the relation
umbers and Irrational numbers.
ch numbers are represented in ascending or
Figure 2: Number Line
ented on the right of the zero and neg
ero as shown in the above diagram.
e information from the real number line into
bols, meaning and representation on the nu
, Meaning and representation on the numbe
Meaning On a nu
not included or open empty c
not included or open empty c
included or closed dense ci
included or closed dense ci
Numbering System
nding and making
phically by a real
hips among Whole
er.
ative numbers are
symbols. Table 1
mber line.
line.
ber line
ircle
ircle
rcle
rcle
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Inequality Notation
Inequalities can be used to describe subsets of real numbers called intervals. An inequality is
a relationship between two unequal quantities and it can be reprented on a number line.
Table 2 show the inequlity notation symbols, meaning and representation on the number
line.
Table 2: Symbol, Meaning and representation on the number line.
Symbol Meaning On a number line
> greater than empty circle
< less than
empty circle
greater than or equal to dense circle
less than or equal to dense circle
Bounded Intervals on the Real Number Line
In the bounded intervals below, the real numbers a and bare the endpoints of each interval.
Interval Notation Inequality Notation Number Line Type
[ ]a,b Closed
[ )a,b Half-open
( ]a,b Half-open
( )a,b Open
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Unbounded Intervals on the Real Number Line
The symbols , positive infinity and , negative infinity do not represent real numbers.
They are simply convenient symbols used to describe the unboundedness of an interval such
as ( )7, or ( ], 4 .
Interval Notation Inequality Notation Number Line Type
( ), Entire Real
line
( )a, Open
[ )a, Half-open
( ), b Open
( ], b Half-open
Example 20
Fill in the blank spaces with the correct answers.
Interval Notation Inequality Notation Number Line
a) ( ),
b) [ )5, 2
c) x 0
d)
2
e) ( ]5, 3
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Interval Notation Inequality Notation Number Line
f)
9
g) x 3<
h) x 4
i) ( ]3,6
j) 2 x 4<
k)
2 3 l) ( )5,12
m) 4 x
n) ( ), 3
o) [ )8, 3
p) 5 x 6
q) [ )7,
r)9 11
s) ( )7,0
t) 5 x
u) [ ]0, 9
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TUTORIAL 2
Fill in the blank spaces with the correct answers.
No. Interval Notation Inequality Notation Number Line
1. x 6
2. ( ]2, 8
3. x 4<
4. ( ]0, 4
5.
7
6. 3 x 6 <
7. [ ]4,12
8. 0 x 8 <
9. x 2
10. ( ]3, 6
11.
3 12. ( )100,
13. 3 x 10 <
14. 6 x
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1.3 Different Form of Numbers
Fraction
A fraction has two parts, known as the numerator and denominator.
4
7
Numerator (the top part)
Denominator (the bottom part)
The denominator shows that the whole been split into 7 parts. The numerator shows 4 parts
of the whole.
Mixed Numbers
A mixed number is a number which consists of a whole number and a fraction.
2 is a whole number4
27
Proper Fraction
Fractions where the numerator is smaller than the denomirator.
4 5 1, ,
7 11 2
Improper Fraction
Fractions where the numerator is equal to or greater than the denominator.
7 15 10, ,7 11 2
Decimal
A decimal is another way of expressing a fraction. The decimal point separates the whole
number from its fractional part. A number written with a decimal point is known as a
decimal.
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Example 21
a) 4 0.410
= b)8
0.08100
= c)12
0.0121000
= d)5 21
2 2.6258 8
= =
Example 22
Fill in the blank spaces below with the correct answers.
Fraction Decimal
a)5
100
b) 0.258
c)4
32
d)0.023
e)13
115
f) 1.54
Percentage
Percentages are expressed as the number of parts in every 100 which means the fraction with
100 as the denominator. The symbol for percentage is %.
Example 23
b) 4 4%100
= b)88
88%100
= c)12.5
12.5%100
= d)0.5
0.5%100
=
and vice versa
Example 24
c)79
79% 100= b)3
3% 100= c)2.5
2.5% 100= d)0.99
0.99% 100=
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A fraction or decimal can be changed into a percentage by multiplying it with 100%.
Fraction/Decimal100%
Percentage
Example 25
Express each of the following values as a percentage.
a) 0.235 b)0.054 c) 1.89 d) 742 e) 100Solution
A percentage can be changed into a fraction or a decimal by dividing it with 100.
Percentage100%
Fraction/Decimal
Example 26
Change the following into a fraction or s decimal
a) 15% b)2.5% c) 297% d) 0.012% e) 100%
Solution
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Standard Form/Scientific Form
We can use standard form to name very large and very small positive numbers and to
perform computations. A number can be written in standard form by expressing it in the
form, nA 10 where 1 A 10 < and n is an integer. The following steps can be used to
express a positive number in standard form.
Step 1: Place a decimal point after the first non-zero digit to obtain the
number A,where 1 A 10 < .
Step 2: Count the number of decimal places between the new point and the
original decimal point. The number gives the value of n.
Step 3: For numbers 10, n is positive and for numbers 1, n< is negative.
Step 4: Write the number in the form nA 10
Example 27
Write the number below in standard form.
a) 0.2004 b) 0.006987 c) 423000 d) 7.22 e) 100Solution
Example 28
Fill in the blanks below.
Fraction Decimal Percentage Standard Form
a) 45.7%
b)3
15
c) 0.052
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Fraction Decimal Percentage Standard Form
d)29
32
e) 16.875 10
f) 2.25
g) 23400%
h) 2754.0
i) 13%
j) 45.0 10
k) 18%
l)3
24
m)3
1
5
n) 0.02
o) 0.044
p)3
325
q) 0.065
r) 1.225%
s) 25.6 10
t) 12.5%
u)2
25
v)27
4000
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TUTORIAL 3
Fill in the blank spaces with the correct answers.
Fraction Decimal Percentage Standard Form
1.9
32
2. 12.34 10
3. 0.015
4. 8.75%
5. 1.2
6. 35%
7. 15.39 10
8. 125%
9.1
14
10.2
25
11. 0.75
12. 1.125
13.3
1100
14. 0.4
15. 103%
16. 21.025 10