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**1.1 INTRODUCTION**

System is a set of principles or procedures according to which something is done.

Number System is a set of principles according to which numbers are represented and various arithmetic operations like addition, subtraction, multiplication etc. are applied on them. This lays down the basic rules on how various digits and symbols are used.

Whenever we change the principles we get a new set of **n****umber s****ystem**. This results in various types of number systems.

**Some of the examples of the number systems are:**

**Base **is defined as the number of digits we use to represent the numbers. **Digits **are the tools which are used to represent numbers.

We will be talking about various number system and the bases in subsequent chapters.

The standard number system that we use across the world is **d****ecimal n****umber s****ystem**. Decimal Number System has a base "10" as it uses 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to represent all the numbers.**1.2 FACE VALUE AND PLACE VALUE****➢ Face Value:** The Value of the digit itself is referred to as the face value of a particular Digit. In decimal number system, we use the 10 digits as mentioned above. Wherever used the face value remains the same.

Example:The face value of 5 is 5 in 568 and 1285 and even 58738.

**➢ Place:** The place value of a number depends on the **p****lace **it is at. The various places and the corresponding multipliers of the places in a decimal number system are listed below. We will be starting from leftmost and move right place by place.**Example:** For a number 786543

**➢ Place value** of a digit in a particular number system is defined as the face value of the number multiplied by the corresponding place. In decimal system the value of the place will be as mentioned in the table above.**Example: **The place value of 5 is 500 in 568 and 5 in 1285 and 50000 in 58738.

Question 1:What is the difference in the two Place Values and Two Face Values of digit 7 in the given number - 78687

In this chapter we are going to talk about the

Numbers, their properties and their classifications. As discussed earlier all these properties will be limited to decimal number system only since that is the basis of all the question asked. The concepts discussed in this chapter will be your first step towards a general understanding of the mathematics requirements to clear any entrance exams. As we proceed you will realize that you have already learnt most of these concept in school.

**1.3 TYPES OF NUMBERS****➢ Natural Number**

The primary application of mathematics is counting. That is where the very first category of numbers comes in i.e. natural numbers or commonly as known as **counting ****Numbers**.

__Features of Natural Numbers:__**(i) **These are the complete numbers i.e. fractions/decimals are not included

**(ii)** Zero is not included**(iii)** Negatives are not included**(iv)** They are represented by "N"**(v) **N = {1, 2, 3, 4, 5.........infinity}

Question 2:Choose the set of Numbers which contains only Natural Numbers

**➢ Whole Numbers**

Natural numbers along with 0, form a set of numbers known as Whole numbers. Since '0' is a representation of absence of anything and is not used for counting, thus they are **not **called the **counting ****numbers**.

__Features of Whole Numbers:__**(i) **These are the complete numbers i.e. fractions/decimals are not included.**(ii) **Zero is included.**(iii)** Negatives are not included.**(iv)** They are represented by "W".**(v)** W = {0, 1, 2, 3, 4, 5.........infinity}.

Question 3:Choose the set of Numbers which contains only Whole Numbers

**➢ Integers**

Numbers less than 'zero' are represented with a negative sign along with them and are referred to as negative numbers.

Whole numbers along with negative of natural numbers form a set of numbers known as integers.__Features of Integers:__**(i) **These are the complete numbers i.e. fractions/decimals are not included.**(ii)** Zero is included.**(iii)** Negatives are included.**(iv) **They are represented by "I".**(v) **I = {- infinity.......-3, -2, -1, 0, 1, 2, 3.........infinity}.

Question 4:Choose the set of Numbers which contains only Integers

**➢ Fraction**

The word ‘fraction’ has been derived from the Latin ‘fractus’ which means “broken”. Fraction represent part of a whole. When we divide a whole into pieces, each part is a fraction of whole.

Example:1/2, 3/5, 2/7, 10/21

A fraction has two parts. The number on the top of the line is called the **numerator**. The number below the line is called the **denominator**.__Features of Integers:__**(i) **These are not complete numbers i.e. integers are excluded.**(ii)** Zero is not a fraction.**(iii) **Fraction can be negative as well as positive.

**Types of Fractions**__Fractions are generally of two types on the basis of representation__:**(i)** Simple Fraction**Example:** 30/7**(ii) **Mixed Fraction**Example:**

Question 5: Convert the mixed fraction into a simple fraction.

Question 6:Convert the simple fraction 112/19 into a mixed fraction.

These are nothing but a form of representation of fractions. We make the value of denominator as unity i.e. "1". They can be both negative and positive.

Example:2.5, 1.25, 1.3333......., 2.666....., 2.8284.....

**Types of Decimal Numbers**

1. __Decimal numbers are majorly divided into two categories__:**(i) ****Terminating Decimal Numbers -** The decimal numbers with a definite end.**Example:** 2.5, 1.25**(ii) Non-Terminating Decimal Numbers -** The decimal numbers without a definite end.**Example:** 1.3333......., 2.666....., 2.8284.....

2. __Non-Terminal Decimal Numbers are further divided into two categories__:**(i) ****Recurring non-terminating decimal numbers**: The non-terminating decimal numbers with digits after the decimal repeating again and again.**(ii) ****Non-recurring non-terminating decimal numbers**: The non-terminating decimal numbers with digits after the decimal do not repeat but have a completely random pattern. These are the irrational numbers which we will discuss ahead in the same chapter.

**➢ Rational Numbers**

All those numbers which can be represented in the form of ^{p}/_{q} where p and q are integers with q ≠ 0 are known as Rational numbers. All the integers, fractions, terminating decimal numbers and non-terminating recurring decimal numbers fall under the category of Rational Numbers.

**➢ Irrational Numbers**

All those numbers which are not rational are known as irrational numbers. Non-reciprocating non-terminating decimal numbers are irrational numbers. Square Roots, Cube Roots, etc of natural numbers turn out to be irrational numbers.**Example:** √3 = 1.7320.... , √8 = 2.8284......, 3√4 = 1.5874.......**♦ Converting a Terminating Decimal Number in ^{p}/_{q }form**

Remove the decimal and divide the number so received by as many 10's as the counting in step 1.

80.125 = 80125/1000 = 641/8

Question 7:Convert 128.8888 in ^{p}/_{q} form

**Example: ****Convert 80. ^{p}/_{q}_{ }form**

x = 80. **Step 2:** Count how many digits are there after decimal, 3 in this case.**Step 3: **Multiply by as many tens on both sides i.e. 1000 in this case.

1000x = 80125. **Step 4:** Subtract from

999x = 80045

x = 80045 / 999**Example:** **Convert 80. 1 ^{p}/_{q}_{ }form**

Let 80.1 = x

Multiply both sides by 100

Subtract from

99x = 7922.4

Question 8:Convert 16. in ^{p}/_{q }form

Question 9:Convert 23.0 in ^{p}/_{q} form

The process we have applied in the examples above is too time consuming we need a method which can do the same process in much less time. For that we have a formula with which you can have the answer in a single step.

Every terminating recurring decimal number has 3 parts (Taking the example of:**(i)** Digits before the decimal (80)**(ii) **No. of digits with bar on them after decimal (2)**(iii) **No. of digits without bar on them after decimal (1)

Rational form =All the digits without decimal written once - All the digits without bar written once / as many 9's as no. of digits with bar on them after decimal followed by as many 0's no. of digits without bar on them after decimal.

**Example: Convert 80. in p/q form****Solution: **

All the digits written once = 80125

All the digits without bar written once = 80

No of digits with bar after decimal = 3

No of digits without bar after decimal = 0

Rational form = (80125 - 80) / 999 = 80045 / 999**Example: Convert 80. 1 in p/q from****Solution: **

All the digits written once = 80125

All the digits without Bar written once = 801

No of digits with bar after decimal = 2

No of digits without bar after decimal = 1

Rational form = (80125 - 801) / 990 = 79224 / 990

Question 10:Convert 16. in ^{p}/_{q} from

Question 11: Convert 23.0in p/q form

Rational numbers, as well as irrational numbers together, form the real numbers. As per the definition, all the numbers which can be represented on a number line are known as real numbers. All the categories which we have discussed till now all fall in the set of Real numbers, including the irrational numbers. They can also be represented on a number lines.

Question 12:Choose the set of Numbers which doesn't contains only Real Numbers

All those numbers which actually do not exist in a number system but we assume their presence for the sake of calculations are known as imaginary numbers. They cannot be represented on number lines. They are also called

Note:The study of complex numbers is not important from Aptitude Testing point of view. So, we will not take this topic any further.

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