MAT 037 - Chapter 1 - Number and Numbering Systems

8/2/2019 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


Number 7 20 23 2

Natural Number

Non-Integer

Number

Even Number

Prime Number


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


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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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



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

MAT 037 - Chapter 1 - Number and Numbering Systems

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