Number
System Aptitude for Competitive Exams

Real Numbers: Real numbers include natural numbers, whole
numbers, integers, rational numbers and irrational numbers.

Natural numbers: 1, 2, 3 …

Whole Numbers: 0, 1, 2, 3 …

Integers : -3, -2, -1, 0, 1, 2, 3 …

Rational Numbers: Rational numbers can be expressed
as ^p/_q where p and q are integers and q≠0
Examples: ¹/₁₂, 42, 0, −⁸/₁₁ etc.

All integers, fractions and terminating or recurring decimals are rational
numbers.How to find rational numbers between two given rational numbers?

If m and n be two rational numbers such
that m < n then 1/2 (m + n) is a rational number between m
and n.

Question: Find out a rational number lying halfway between 2/7 and 3/4.

Solution:

Required number = 1/2 (2/7 + 3/4) 

= 1/2 ((8 + 21)/28) 

= {1/2 × 29/28) 

= 29/56

Hence, 29/56 is a rational number lying halfway between 2/7 and 3/4.

Question: Find out ten rational numbers
lying between -3/11 and 8/11.

Solution:

We know that -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 <
6 < 7 < 8

Therefore, -3 /11< -2/11 < -1/11 < 0/11 < 1/11 < 2/11 < 3/11
< 4/11 < 5/11 < 6/11 < 7/11 < 8/11

Hence, -2/11, -1/11, 0/11, 1/11, 2/11, 3/11, 4/11, 5/11, 6/11 and 7/11 are the
ten rational numbers lying between -3/11 and 8/11.

Irrational Numbers

Any number which is not a rational
number is an irrational number. In other words, an irrational number is a
number which cannot be expressed as p/q, where p and q are integers.

For instance, numbers whose decimals do not terminate and do not repeat cannot
be written as a fraction and hence they are irrational numbers.

Example: π, √2, (3+√5), 4√3 (meaning 4×√3), 6√3 etc

Please note that the value of π = 3.14159 26535 89793 23846
26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280
34825 34211 70679…

We cannot write π as a simple fraction (The fraction ²²/₇ =
3.14…. is just an approximate value of π)

Complex Numbers:

A complex number is a number that can
be expressed in the form a + bi, where a and b are real numbers and i is the
imaginary unit, that satisfies the equation i² = −1. In this expression, a is the real
part and b is the imaginary part of the complex number.

Even NNumbers: Divisible by 2 (2, 4, 6, 8, 22, 44, 68, 34234… )

Odd NNumbers: NOT divisible by 2 (3, 5, 7, 9, 23, 45, 67, 34235… )

Prime Numbers: 
A number greater than 1 is called a
prime number, if it has only two factors, namely 1 and the number itself.

Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Procedure to find out the prime number

Suppose A is the given number.

Step 1: Find a whole number nearly greater than the square root of A. 
Let K is nearby square root of A 
Step 2: Test whether A is
divisible by any prime number less than K. If yes A is not a prime number. If
not, A is a prime number.

Example:

Find out whether 337 is a prime number or not?

Step 1: 19 is the nearby square root (337) Prime numbers less than 19 are 2, 3,
5, 7, 11, 13, 17 
Step 2: 337 is not divisible by any of
them

Therefore, 337 is a prime number

Co-Prime Numbers: 

In number theory, two integers a and b
are said to be co-prime if the only positive integer that evenly
divides both of them is 1. That is, the only common positive factor of the two
numbers is 1. This is equivalent to their H.C.F. being 1. e.g. (2,3), (6,13),
(10,11), (25,36) etc

Composite Numbers:

A natural number greater than 1 that is
not a prime number is called a composite number. e.g. 4, 6, 8, 9, 10, 12 etc

Perfect Numbers:

A perfect number is a positive integer that is equal to the sum of its positive
divisors excluding the number itself (proper positive divisors). 

The first perfect number is 6, because 1, 2, and 3 are its
proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is
equal to half the sum of all its positive divisors: ^{( 1 + 2 + 3 + 6 )}/₂=
6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14.

There are total 27 perfect numbers.

Surds :

Let a be any rational number and n be
any positive integer such that a√n is irrational. Then a√n is a surd.

Example: √3, 10√6, √43 etc

Every surd is an irrational number. But every irrational number is
not a surd. (E.g.: Π, e etc are not surds though they are irrational
numbers.)

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