SIMPLIFICATION

BASIC FORMULA

1. (a+b) 2=a2+b2+2ab 
    
2. (a−b) 2=a2+b2−2ab

3. (a +b) 2− (a−b) 2=4ab  
                      
4. (a+b) 2+ (a−b) 2=2(a2+b2)

5. (a2–b2)= (a+b) (a−b) 
                   
6. (a+b+c) 2=a2+b2+c2+2(ab+bc+ca)

7. (a3+b3) = (a+b) (a2−ab+b2)   

8. (a3–b3) = (a−b) (a2+ab+b2)

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

10. If a+b+c=0, then a3+b3+c3=3abc.


TYPES OF NUMBERS

1. Natural Numbers:
Counting numbers 1, 2, 3, 4, 5 … are called natural numbers

2. Whole Numbers:
All counting numbers together with zero form the set of whole numbers.
Thus,
(I) 0 is the only whole number which is not a natural number.
(II) Every natural number is a whole number.

3. Integers:
  All  natural  numbers,  0  and  negatives  of  counting  numbers i.e.,…,−3,−2,−1,0,1,2,3,….. together form the set of integers.
(i) Positive Integers: 1, 2, 3, 4….. is the set of all positive integers.
(ii) Negative Integers: −1, −2, −3… is the set of all negative integers.
(iii) Non-Positive and Non-Negative Integers: 0 is neither positive nor negative.
So,  0,1,2,3,….  represents  the  set  of  non-negative  integers, 
while 0,−1,−2,−3,….. represents the set of non-positive integers.

4. Even Numbers:
A number divisible by 2 is called an even number, ex. 2, 4, 6, 8, etc.

5. Odd Numbers:
A number not divisible by 2 is called an odd number. e.g. 1, 3, 5, 7, 9, 11 etc.

6. Prime Numbers:
A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.   e.g  2

7. Composite Numbers:
Numbers  greater  than  1  which  are  not  prime,  are  known  as composite numbers, e.g., 4,6,8,9,10,12.
Note:
(i) 1 is neither prime nor composite.
(ii) 2 is the only even number which is prime.
(iii) There are 25 prime numbers between 1 and 100.
REMAINDER AND QUOTIENT:

"The remainder is r when p is divided by k" means p=kq+r the integer q is called the quotient.

EVEN ,ODD NUMBERS

A number n is even if the remainder is zero when n is divided by 2: n=2z+ 0 or n=2z.
A number n is odd if the remainder is one when n is divided by 2: n=2z+1.
even X even = even
odd X odd = odd
even X odd = even
even + even = even
odd + odd = even
even + odd = odd



                                  Some important tricks
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)

                    For square and square root or cube and cube root
You must remind square from number 1 to 30  and cube at least from number 1 to 15

Square List                                                                                                   Cube List 
 

1 1 1 1 2 4 2 8 3 9 3 27 4 16 4 64 5 25 5 125 6 36 6 216 7 49 7 343 8 64 8 512 9 81 9 729 10 100 10 1000 11 121 11 1331 12 144 12 1728 13 169 13 2197 14 196 14 2744 15 225 15 3375 16 256 17 289 18 324 19 361 20 400 21 441 22 484 23 529 24 576 25 625 26 676 27 729 28 784 29 841 30 900
Also  You must remind  to  1 to 20  table multiplication
TABLES (from 1 TO 20) 1x1=1 2xl=2 3x1=3 4xl=4 5xl=5 1x2=2 2x2=4 3x2=6 4x2=8 5x2=10 1x3=3 2x3=6 3x3=9 4x3=12 5x3=15 1x4=4 2x4=8 3x4=12 4x4=16 5x4=20 1x5=5 2x5=10 3x5=15 4x5=20 5x5=25 1x6=6 2x6=12 3x6=18 4x6=24 5x6=30 1x7=7 2x7=14 3x7=21 4x7=28 5x7=35 1x8=8 2x8=16 3x8=24 4x8=32 5x8=40 1x9=9 2x9=18 3x9=27 4x9=36 5x9=45 1x10=10 2xlO=20 3xlO=30 4x10=40 5xlO=20 6x1=6 7x1=7 8x1=8 9xl=9 lOx1=10 6x2=12 7x2=14 8x2=16 9x2=18 10x2=20 6x3=18 7x3=21 8x3=24 9x3=27 10x3=30 6x4=24 7x4=28 8x4=32 9x4=36 10x4=40 6x5=30 7x5=35 8x5=40 9x5=45 10x5=50 6x6=36 7x6=42 8x6=48 9x6=54 10x6=60 6x7=42 7x7=49 8x7=56 9x7=63 10x7=70 6x8=48 7x8=56 8x8=64 9x8=72 10x8=80 6x9=54 7x9=63 8x9=72 9x9=81 lOx9=90 6x10=60 7x10=70 8x10=80 9xlO=90 10x10=100 11x1=11 12xl=12 13x1=13 14x1=14 15x1=15 11x2=22 12x2=24 13x2=26 14x2=28 15x2=30 11x3=33 12x3=36 13x3=39 14x3=42 15x3=45 11x4=44 12x4=48 13x4=52 14x4=56 15x4=60 11x5=55 12x5=60 13x5=65 14x5=70 15x5=75 11x6=66 12x6=72 13x6=78 14x6=84 15x6=90 11x7=77 12x7=84 13x7=91 14 x7=98 15x7=105 11x8=88 12x8=96 13x8=104 14x8=112 15x8=120 11x9=99 12x9=108 13x9=117 14x9=126 15x9=135 11xl0=110 12xl0=120 13xl0=130 14xlO=140 15xl0=150 16x1=16 17x1=17 18x1=18 19x1=19 20x1=20 16x2=32 17x2=34 18x2=36 19x2=38 20x2=40 16x3=48 17x3=51 18x3=54 19x3=57 20x3=60 16x4=64 17x4=68 18x4=72 19x4=76 20x4=80 16x5=80 17x5=85 18x5=90 19x5=95 20x5=100 16x6=96 17x6=102 18x6=108 19x6=114 20x6=120 16x7=112 17x7=119 18x7=126 19x7=133 20x7=140 16x8=128 17x8=136 18x8=144 19x8=152 20x8=160 16x9=144 17x9=153 18x9=162 19x9=171 20x9=180 16xl0=160 17xlO=170 18x10=180 19x1O=190 20x10=200