MAT 107 Number,Chapter-1

Number

From time immemorial human beings have been trying to have a count of their

belongings- goods, ornaments, jewels, animals, trees, sheeps/goats, etc. by using various

techniques

- putting scratches on the ground/stones

- by storing stones - one for each commodity kept/taken out.

This was the way of having a count of their belongings without having any knowledge of

counting.

One of the greatest inventions in the history of civilization is the creation of numbers. You can

imagine the confusion when there were no answers to questions of the type “How many?”,

“How much?” and the like in the absence of the knowledge of numbers. The invention of

number system including zero and the rules for combining them helped people to reply

questions of the type:

(i) How many apples are there in the basket?

(ii) How many speakers have been invited for addressing the meeting?

(iii) What is the number of toys on the table?

(iv)How many bags of wheat have been the yield from the field?

The answers to all these situations and many more involve the knowledge of numbers and

operations on them. This points out to the need of study of number system and its extensions

in the curriculum. In this lesson, we will present a brief review of natural numbers, whole

numbers and integers. We shall then introduce you about rational and irrational numbers in

detail. We shall end the lesson after discussing about real numbers.

Since numbers are such an integral part of our lives, we tend to take them for granted. When

we learned to count , we had the names and symbols for counting numbers at our disposal

from the beginning.

As we encountered measurement, we often found that the object being measured did not

contain the unit of measurement a whole numbers of times. This necessitated the use of

fractional parts of a unit. Going one step further, we also recognized that the measure of

certain quantities could not represented exactly, even by using fractional parts of units. So

again new types of numbers arose. In this chapter we will consider various types numbers and

common operations on those

Natural numbers , Whole Numbers:

The number which we count , that is ,the numbers 1,2,3 etc are called the natural numbers. We

will denote the collection or set of natural numbers by N .If we include the number 0 along

with the natural number , we get the set of whole numbers , which is denoted by W .

Since the set of natural numbers and whole numbers contain infinitely many number we can ‘t

list actually every number .

Definition :

Whole number: The number 0,1,2,3,……………… are called the whole numbers.

Natural number: The number 1,2,3,……………… are called the natural numbers.

Number line:

We use a number line to represent the number system. We draw a straight line , usually in a

horizontal or vertical position. We fix an arbitrary point anywhere on the number line and

named as origin and assign it the number 0.We then decide on the distance that we will use for

the length of one unit on the number line and the direction that will be the positive direction on

the line. The number 1 is assigned to the point located one unit length in the positive direction

from the origin. Likewise the number 2 is two unit distance in the positive direction from the

origin. Continuing in this way we have others .similarly we can take -1, -2 in the negative

direction.

Note the following properties of the whole numbers 0 and 1

1. aa 0 for any other number a .Because of this property we call 0 the additive identity.

2. aa 1. for any other number a .Because of this property we call 0 the multiplicative

identity.

3. 00

bif 0 is being divided by a nonzero number , the quotient is zero.

4. 0

ais not defined. It is impossible

Integer

While dealing with natural numbers and whole numbers we found that it is not always possible

to subtract a number from another.

For example, (2 – 3), (3 – 7), (9 – 20) etc. are all not possible in the system of natural numbers

and whole numbers. That is when a number is subtracted from another small number. Thus, it

needed another extension of numbers which allow such subtractions. We introduce this type of

number as negative integer. The set of integer number is denoted by I .

Absolute value of a number:

The absolute value of a number is the distance of the number from the origin of the number

line

Example :Since the distance from the origin to the point corresponding to -4 is 4 units, the

absolute value of -4 is 4, denoted by 4 ,is 4, ie. 44

The absolute value of a number is always greater than or equal to zero..

In number line

"6" is 6 away from zero,

and "−6" is also 6 away from zero.

So the absolute value of 6 is 6,

and the absolute value of −6 is also 6

Definition

…….-4 ,-3,-2.-1,0,1,2,3,4…………… are called the integer number.

Rational number

Consider the situation, when an integer a is divided by another non-zero integer b .

Definition

A rational number is a number that can be written in the form

b

a

Where a and b are the integer and 0b

The set of rational number is denoted by Q and is defined by also in the following way

0,,, bIbab

aQ

Example:

1,5,5.3,

4

1

A rational number is a number that can be written as a decimal that either terminates or

repeats .

By terminating decimal , we mean one that eventually stops; it has only a finite number of

nonzero decimal places. For example 1.25, -0.2587.

Repeating decimal is one that continues on forever but eventually begins to repeat the same

digit or block of digits over and over without end . Examples of repeating decimals are

0.666….,2.08333……., -0.8989……, -5.4123123…..,The dots in the preceding numbers indicate

that the repeating digit or block of digits continues to repeat forever.

Each rational number is represented by a point on the number line.

Irrational Number :

An irrational number is any real number that cannot be expressed as a ratio of integer.

Informally, this means that an irrational number cannot be represented as a simple

fraction. Irrational numbers are those real numbers that cannot be represented as

terminating or repeating decimals.

The set of irrational number is denoted by /Q

Definition :

Any number that can’t be possible to express as b

aform is called an irrational number.

Where a and b are the integer and 0b

Example:

,6,2

Real numbers:

When we combine the rational numbers with the irrational numbers, we form a bigger

collection of numbers called the set of real numbers. It is denoted by R.

Definition:

All the numbers rational and irrational together are called the real numbers.

Example:

2

1,4,1,,6,2

The real numbers fill up the number line.

The square root of a given number is the positive number that when multiplied by itself, yields

the given number as the product. the symbol used to denote a root of number is called

radical sign . the number under the radical sign is called the radicand. We say that the radicand

is a perfect square if it can be written as some rational number. That rational number is then

the square root of the radicand. That the square root operation undoes what the squaring

operations did

Example

416 since 1644

3

2

9

4 since

9

4

3

2

3

2

Again 5,2 are irrational number since 2,5 are not perfect square.

Prime Number:

A prime number is not expressible as a product of factors other than 1 and itself.

The prime numbers between 1 to 20 are 2,3,5,7,11,13,17 and 19.

Composite number :

A composite number is expressible as a product of prime factors other than 1 and itself.

The prime factorization of a composite number is the factorization of the number as a product

of prime factors .

Example : 84=2.2.3.7

Imaginary Number:

An imaginary number is a number that can be written as a real number multiplied by

the imaginary unit i, which is defined by its property i2 = −1 .The square of an imaginary

number bi is −b2. For example, 5i is an imaginary number, and its square is −25. Except

for 0 (which is both real and imaginary), imaginary numbers produce negative real

numbers when squared.

Complex Number:

If a real number is added to an imaginary number then the new number is called the

complex number. The set of complex number is denoted by C .It can be defined in the

following way

Definition:

Any number is of the form iba is called the complex number where a is the real part

and bi is imaginary part.

Example: .21.32 ii

Fraction in lowest Term

A fraction in lowest terms if there are no common factors in the numerator and

denominator.

Example 3

2

18

12

Exercise

1. Define Natural number, integer number rational number, irrational number, real number,

Prime Number.

2. Identify the following numbers as rational, irrational, or real. Numbers will fall into more than

one category.

(i). a. 2 b. 2 C. 2

1 d. e. 0 f. 6

(ii). a. -5 b. 3 C. 4.285 d. e. 3.14

3. Evaluate the following square roots of perfect squares:

(i). a. 64 b. 100 C. 4

1 d.

16

9 e. 01.0 f. 49.0

(ii). a. 36 b. 144 C. 25

1 d.

49

81 e. 09.0 f. 21.1

4. Identify each of the following square roots as rational, irrational, or nonreal.

(i). a. 3

2 b.

25

4 C. 1 d. 11 e. 4 f. 03.0

(ii). a. 01.0 b. 4

3 C. 16 d. 64 e. 9.9 f. 2.1

Scientific Notation:

Scientific notation (also referred to as "standard form" or "standard index form") is a

way of writing numbers that are too big or too small to be conveniently written in decimal

form. Scientific notation has a number of useful properties and is commonly used in

calculators and by scientists, mathematicians and engineers.

In scientific notation all numbers are written in the form

Example: it is easier to write (and read) 1.3 × 10-9 than 0.0000000013

Definition: A number is written in scientific notation if it is expressed as the product of a power

of ten and a decimal number with one nonzero digit to the left of the decimal point.

Example : 23 107.8,1097.3

1. The decimal point in a whole number is understood to follow the last digit. We must move it

left until it follows the first(leftmost) digit. Then we multiply by the power of ten that

corresponds to the number of places the decimal point has moved.

Example: 4103219.5219,53

2. In a decimal number the decimal point position is known. We simply move it left or right until

it follows the leftmost nonzero digit. Then we multiply by the power of ten that corresponds to

the number of places moved. Left moves require positive powers and right moves require

negative powers.

Example:

110537.437.45

310146.7007146.0

3.If the number is in power of ten notation with one nonzero digit left to the decimal point , the

it is in scientific notation. If not, we deal with the number as in case 1 and 2(whichever applies)

and change the power of ten by addition or subtraction of the number of places the decimal

point has been moved.

Example:

43 105.51055

23 105.41045

Exercise

1. Write the following numbers in scientific notation.

(i). a. 9563 b. 5.49 C. 0.0362

d. 310891 e. 11018.0 f. 41002.76

(ii). a. 325,000 b. 2.678 C. 0.82

d. 31064 e. 2100055.0 f. 6102184

2. Perform the following operations on numbers written in scientific notation. Write your

answers in scientific notation and as ordinary numbers.

a. 33 107.6108.4

b. 44 1082.5103.5

c. 66 10356.11063.7

d. )103.8()1075.3( 33

e. )104.6()10397.1( 2

f. )1008.5()1054.2( 3

g. )102.3()10152.3( 79

2. Perform the following operations on numbers written in scientific notation. Write your

answers in scientific notation and as ordinary numbers.

a. 44 1084.7108.28.9

b. 33 1075.51096.2

c. 65 1039.1101.6

d. )10943.4()107( 72

e. )103.8()1085.9( 23

f. )103()10729.6( 32

g. )1036.6()10406.5( 64

Summary:

Natural number : one of the numbers 1,2,3,4,……………..

Whole number: one of the numbers 0,1,2,3,4,……………..

Number line : a line on which we graphically represent numbers by points.

Origin Of number line :the point on number line corresponding to the number 0.

Prime number : a natural number that is not expressible as product of prime factors other than

and itself.

Composite Number: a number that can be written as a product of prime factors other than 1

and itself.

Integer: one of the numbers…………-3,-2,-1,0, 1,2,3,4,……………..

Absolute value of a number: the distance of the number from the origin of the number line ; it

is never negative.

Rational number : a rational number is a number that can be written in the form

b

a

Where a and b are the integer and 0b

Irrational number : a number that is not rational.

Real number: either a rational or an irrational number.

Square root of a number: that positive number which , when multiplied by itself , yields the

given number.

Radical sign : the symbol used to denote a root of number