Important Notes on Number System..!

Important Notes on Number System..!

The number system is an important topic for upcoming SSC & Railways Exams. Here, we are going to help you with Basic Concepts & Short Tricks on Number System in Quant Section. We will be providing you with details of the topic to make the Quant Section and calculation easier for you all to understand.

1.Natural Numbers: Numbers which are used for counting the objects are called natural numbers. They are denoted by N.

N = { 1, 2, 3………………..}

All positive integers are natural numbers.

2.Whole numbers: When  ‘zero’ is included in the natural numbers, they are known as whole numbers.

They are denoted by W.

W= { 0, 1, 2, 3……………….}

3. Integers: All natural numbers, zero and negatives of natural numbers are called as integers.

They are denoted by I.

I = { ………………..,-3, -2, -1, 0, 1 , 2, 3………………}

4. Rational numbers: The numbers which can be expressed in the form of   where P and Q are integers and  are called rational numbers

They are called by Q.

 

5. Irrational numbers: The numbers which cannot  be written in the form of  where P and Q are integers and  are called irrational numbers.

 

When these numbers are expressed in decimal form, they are neither terminating nor repeating.

6. Real numbers: Real numbers include both rational as well as irrational numbers.

Positive or negative, large or small, whole numbers or decimal numbers are all real numbers.

e.g.= 1,    13.79,   -0.01,   etc.

7. Imaginary numbers: An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit ‘i’ which is defined by its properly 

Note: Zero (0) is considered to be both a real and imaginary numbers.

8. Prime number: A prime number is a natural number greater than 1 and is divisible only by 1 and itself.

e.g.2, 3, 5, 7, 11, 13, 17 ,19 ………….etc.

Note:- 2 is the only even prime number.

9. Composite Numbers: A number, other than 1, which is not a prime number is called a composite number.

E.g. 4, 6, 8, 9, 10, 12, 14, 15 ……….etc.

Note :1   1 is neither a prime number nor a composite number.

            2 there are 25 prime numbers between 1 and 100.

10.Complex Number: The numbers formed by the combination of real numbers and imaginary numbers are called the complex number. Every complex number is written in the following form:

A+iB, where A is the real part of the number and B is the imaginary part.

11. Even & Odd Numbers: All the numbers divided by 2 are even numbers. Whereas the ones not divisible by 2 are odd numbers.

Example: 4, 6, 64, 100, 10004, etc are all even numbers.

13Perfect Numbers: All the numbers are called perfect numbers if the sum of all the factors of that number, excluding the number itself and including 1, equalizes the to the number itself then the number is termed as a perfect number. 

Example:6 is a perfect number. As the factors of 6= 2 and 3. 

As per the rule of perfect numbers, sum= 2+3+1 = 6. Hence, 6 is a perfect number.

Some important properties of Numbers:

The number 1 is neither prime nor composite.

The only number which is even is 2.

All the prime numbers greater than 3 can be written in the form of (6k+1) or (6k-1) where k is an integer.

Square of every natural number can be written in the form 3n or (3n+1) and 4n or (4n+1).

The tens digit of every perfect square is even unless the square is ending in 6 in which case the tens digit is odd.

The product of n consecutive natural numbers is always divisible by n!, where n!= 1X2X3X4X….Xn (known as factorial n).

Number System Formulas:

1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2

(12 + 22 + 32 + ….. + n2) = n ( n + 1 ) (2n + 1) / 6

(13 + 23 + 33 + ….. + n3) = (n(n + 1)/ 2)2

Sum of first n odd numbers = n2

Sum of first n even numbers = n (n + 1)

(a + b)(a – b) = (a2 – b2)

(a + b)2 = (a2 + b2 + 2ab)

(a – b)2 = (a2 + b2 – 2ab)

(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

(a3 + b3) = (a + b)(a2 – ab + b2)

(a3 – b33) = (a – b)(a2 + ab + b2)

(a3 + b3 + c3 – 3abc) = (a + b + c)(a2 + b2 + c2 – ab – bc – ac)

When a + b + c = 0, then a3 + b3 + c3 = 3abc

Divisibility Rules:

A number is divisible by 2 if its unit’s digit is any of 0, 2, 4, 6, 8.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 4 if the number formed by the last two digits is divisible by 4.

A number is divisible by 5 if its unit’s digit is either 0 or 5.

A number is divisible by 6 if it is divisible by both 2 and 3.

A number is divisible by 8 if the number formed by the last three digits of the given number is divisible by 8.

A number is divisible by 9 if the sum of its digits is divisible by 9.

A number is divisible by 10 if it ends with 0.

A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11.

A number is divisible by 12 if it is divisible by both 4 and 3.

A number is divisible by 14 if it is divisible by 2 as well as 7.

Two numbers are said to be co-prime if their H.C.F. is 1. To find if a number, say y is divisible by x, find m and n such that m * n = x and m and n are co-prime numbers. If y is divisible by both m and n then it is divisible by x.

Example:

(1) What is the total of all the even numbers from 1 to 400?

Solution:

From 1 to 400, there are 400 numbers. So, there are 400/2= 200 even numbers.

Hence, sum = 200(200+1) = 40200     (From Rule III)

(2) What is the total of all the even numbers from 1 to 361?

Solution:

From 1 to 361, there are 361, there are 361 numbers; so there are even numbers. Thus, sum = 180(180+1)=32580

(3) What is the total of all the odd numbers from 1 to 180?

Solution:

Therefore are 180/2 = 90 odd numbers between the given range. So, the sum =

(4) What is the total of all the odd numbers from 1 to 51?

Solution

There are odd numbers between the given range. So, the sum =

(5) Find the of all the odd numbers from 20 to 101.

Solution:

The required sum = Sum of all the odd numbers from 1 to 101.

Sum of all the odd numbers from 1 to 20

= Sum of first 51 odd numbers – Sum of first 10 odd numbers

=

Miscellaneous

1. In a division sum, we have four quantities – Dividend, Divisor, Quotient and Remainder. These are connected by the relation.

Dividend = (Divisor × Quotient) + Remainder

2. When the division is exact, the remainder is zero (0). In this case, the above relation becomes

Dividend = Divisor × Quotient

Example: 1: The quotient arising from the divisor of 24446 by a certain number is 79 and the remainder is 35; what is the divisor?

Solution:

Divisor × Quotient = Dividend -  Remainder

79×Divisor = 24446 -35 =24411

Divisor = 24411 ÷ 79 = 309.

Example: 2: A number when divided by 12 leaves a remainder 7. What remainder will be obtained by dividing the same number by 7?

Solution:

We see that in the above example, the first divisor 12 is not a multiple of the second divisor 7. Now, we take the two numbers 139 and 151, which when divided by 12, leave 7 as the remainder. But when we divide the above two numbers by 7, we get the respective remainder as 6 and 4. Thus, we conclude that the question is wrong.