Real Numbers
Numbers On line... Its set of numbers contains...
#Natural Numbers
#Whole Numbers
#Prime Numbers
#Composite Numbers
#Rational_&_Irrational Numbers... also🤘

Encircled are #Prime_numbers Other are #Composites Except '1'...

Compex Numbers
Numbers expressed in the form a + bi where a & b are real Numbers & #_i is imaginary number (i.e. i=√(-1) )...

Even Numbers
Numbers Divisible by 2...
Eg. 4, 6, 8, 10,......

Odd Numbers
Numbers not Divisible by 2
Eg. 1, 3, 5, 7,...

Operations...

1.Addition of two numbers...

1. (+a)+(+b)=+ve
Eg. (2+3)=+5

2. (-a)+(+b)= -ve/+ve...sign of greater number...
Eg. (-3)+(+4)=(-3+4)=(+1)

3. (+a)+(-b)= (a-b) = -ve/+ve... Sign of greater number...
Eg. (+5)+(-7)=(5-7)=(-2)

4. (-a)+(-b)=(-a-b)= -ve...
Eg. (-3)+(-2)= -(3+2)=-(5)
Or = (-3-2)=(-5) #same


2. Substraction of two numbers

1. (+a)-(+b)= +ve If_(a>b) & -ve If_(a<b)
Eg. (+2)-(+6)=(2-6)=(-4)

2. (+a)-(-b)=(a+b)=+ve
Eg. (6-(-3)= (6+3) =+9

3. (-a)-(-b)=(-a+b)= Sign of greater number
Eg. (-2)-(-5)=-(2-5)=-(-3)=3

4. (-a)-(+b)=(-a-b)=-ve(a+b)
eg. (-5)-(+3)=(-5-3)=-(5+3)= -(8)
Or =(-8)... #Same

3.Multiplication

1. (+a)×(+b)=+be
Eg. (8×2)=+16
i.e.Product of two positive numbers carries positive number.

2. (+a)×(-b)=-ve
Eg. (6×(-4))=(-24)

3. (-a)×(+b)=-ve
Eg. (-7)×(+5)=(-35)
⏫ Product of two numbers must be negative iff one of the number is Negative.

4. (-a)×(-b)=+ve
Eg. (-8)×(-2)= +16
i.e Product of two Negative numbers carries Positive Number.


4. Division

1. (+a)÷(+b)=+ve
Eg. (+1)÷(+4)= +0.25

2. (+a)÷(-b)= -ve
Eg. (+2)÷(-4)= -0.5

3. (-a)÷(+b)= -ve
Eg. (-4)÷(+4)= -1

4. (-a)÷(-b)= +ve
Eg. (-6)÷(-3)= +2

( Division of two numbers must be positive...iff both the numbers are either positive or negative.)

( Division of two numbers will be equal to Negative... If n only if one of the number is Negative.)

5. Exponentiation
i.e Numbers in the form bn or b^n...
We can say 'b to the power n'.


1. (+a)^n=+ve
Eg. (+5)^3=5×5×5= (+125)

2. (-a)^n= +ve Or -ve
#depends on the value of 'n'... If 'n' is Even result will be Positive.
If 'n' is Odd result will be Negative.

Eg.1.Even power (-4)^4=(-4)×(-4)×(-4)×(-4)=(+128)

2.Odd power
(-5)^3=(-5)×(-5)×(-5)=(-125)

Simplification of Expressions

Order of Operations in Maths...#PEMDAS
BODMAS

1.Parenthesis i.e Bracket(),[],{} first
From innermost to outer

2. Exponent i.e power (like square)

3. Multiplication & Division (left to right)

4. Addition & Substraction (left to right)

5.Distribute when brackets exist.
Example:
2(x + y) + 3(x - y)
= 2x + 2y + 3x - 3y
= 5x - y

Example #order of operations...

Eg. 1: 6+(14-3)×5-5²+22÷2=??
Ans:-
Solve Bracket first,
=6+(11)×5-5²+22÷2
Now,Exponent
=6+11×5-25+22÷2
Now,Multiply&Divide from left to right,
=6+55-25+11
Now, Addition & Subtraction from left to right,
=61-25+11
=36+11
=47...!

Divisibility Rules 👆

Important formulas

1. a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca)

2. (a+b)³= a³+3a²b+3ab²+b³

3. a³+b³= (a+b)(a²-ab+b²)

4. a³-b³= (a-b)(a²+ab+b²)

5. (a+b+c)²= a²+b²+c²+2ab+2bc+2ca

6. (a+b)²= a²+2ab+b²

7. (a-b)²= a²-2ab+b²

8. a²-b²= (a-b)(a+b)
...

You should know the tables from 1-25...!

Assuming that, We know square of each number from 1-25. We are going to find square of of any number bigger than 25. By assuming base as 50, 100, 150, 200, 250, 300......

Base 50 : i.e for numbers between 26-75.
1. Get the difference between 50 and given number and square it(Write it on right side such that it will consume last two digits.
2. Subtract the difference( if number < 50) or Add the difference (if number > 50) in 25. And Write result in front of result of case 1.

e.g (46)^2 = (25-4)=21 and (4^2)=16 i.e 2116

Same for 57^2 = (25+7) and (7^2)
i.e 3249

Base 100 : i.e for numbers between 76-125.
1. Get the difference between 100 and given number and square diff(Write it on right side such that it will consume last two digits and carry 100th digit if 3digits are there.
2. Subtract the difference( if number < 100) or Add the difference (if number > 100) in given number And Write result in front of result of case 1.

Example 1: 104 × 104
diff- 4... Square - 16
104+4=108
Answer: 10816

Example 2: 97 × 97
Case1- Diff- 3 square 09
Case2- 97-3 = 94
Concate case1 and case2
Answer: 9409