We will be talking about various number systems and the bases in subsequent chapters. The standard number system that we use across the world is the decimal number system. 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.
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.
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
Natural numbers along with 0, form a set of numbers known as Whole numbers. Since '0' is a representation of the absence of anything and is not used for counting, thus they are not called the counting numbers.
Features of Whole Numbers
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
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 the 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.
Types of Fractions
Fractions are generally of two types on the basis of representation:
(i) Simple Fraction
Example: 30/7
(ii) Mixed Fraction
Example:
These are nothing but a form of representation of fractions. We make the value of the 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
Decimal numbers are majorly divided into two categories:
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 nonterminating recurring decimal numbers fall under the category of Rational Numbers.
Classification of Rational Numbers
All those numbers which are not rational are known as irrational numbers. Nonreciprocating nonterminating 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:
Example: Convert 80.125 in ^{p}/_{q}_{ }form
 Count how many digits are thereafter decimal, 3 in this case.
 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
Example: Convert 80. in ^{p}/_{q}_{ }form.
 Let the number be equal to x
⇒ x = 80. Count how many digits are thereafter decimal, 3 in this case.
 Multiply by as many tens on both sides, i.e. 1000, in this case.
⇒ 1000x = 80125. Subtract from
⇒ 999x = 80045
⇒ x = 80045 / 999
Example: Convert 80. 1 in ^{p}/_{q}_{ }form.
 Let 80.1 = x
 Multiply both sides by 100
⇒ Subtract from
⇒ 99x = 7922.4
The process we have applied in the examples above is too timeconsuming. We need a method that 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 the bar on them after the decimal (2).
(iii) No. of digits without a bar on them after the 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
 All the digits are 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
 All the digits are 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 = 79324 / 990
Note: The study of complex numbers is not important from Aptitude Testing point of view. So, we will not take this topic any further.
Certain Operations on Whole Numbers:
(i) Odd ± Odd = Even
(ii) Even ± Even = Even
(iii) Odd ± Even = Odd
(iv) Even ± Even ± Even ........... n terms = Even
(v) Odd ± Odd ± Odd .............. n terms = Even if n is Even, Odd if n is Odd
(vi) Odd × Odd = Odd
(vii) Even × Odd = Even
(viii) Even × Even = Even
(ix) Even × Even × Even ....... n terms = Even
(x) Odd × Odd × Odd ....... n terms = Odd
(xi) (Even)^{n} = Even
(xii) (odd)^{n} = Odd
(a) Prime Numbers: Being Prime is a property of Natural Numbers. All those numbers which are divisible by exactly 2 numbers, i.e. itself and unity "1".
Example: 2, 3, 5, 7, 11 etc.
(b) Composite are the numbers which are divisible by more than 2 numbers, i.e. itself, unity "1" and at least one other.
Example: 4, 6, 8, 9, 10 etc.
1 is neither Prime nor Composite.
➢ Property of Prime Numbers
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Important Concepts: Number System  2 Doc  8 pages 
Practice Test: Number System 1 Test  20 ques 
Practice Test: Number System 2 Test  20 ques 
1. What is the difference between face value and place value in the number system? 
2. What are the different types of numbers in the number system? 
3. How can understanding the number system concepts help in solving mathematical problems? 
4. How is the concept of place value important in understanding the decimal system? 
5. Why is it essential to differentiate between face value and place value when working with numbers? 
194 videos156 docs192 tests

Important Concepts: Number System  2 Doc  8 pages 
Practice Test: Number System 1 Test  20 ques 
Practice Test: Number System 2 Test  20 ques 

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