SIMPLIFICATION

SIMPLIFICATION

BASIC FORMULAE

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.

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

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

3. (13 + 23 + 33 + ..... + n3) = (n(n + 1)/ 2)2

4. Sum of first n odd numbers = n2

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

For square and square root or cube and cube root

You must learn square and cube at least from number 1

to 50

Short tricks of multiplication

Multiplication of Two digit numbers:

(1) 13 x 13 = ?

The result of multiplication of two digit number is 13×13 =

169.

Step 1: Multiply 3×3 = 9 then,

Step 2: Do Cross-multiplication (1×3) = 3 and (1×3) = 3.

Step 3: Add both the result (1×3 + 1×3) = 6 and write

down to the left of 9 (result of step 1).

Step 4: Multiply left hand side numbers (1×1) = 1 and

write down to the left of 6 (result of step 3).

Finally the result we get 169. (Try to calculate all four

steps in mind.)

(2) 87 x 33 =?

The result of multiplication of two digit number is 88 x 33

= 2871

Step 1: Multiply (3×7) = 21 note down 1 and carry 2 then,

Step 2: Do Cross-multiplication (3×8) = 24 and (3×7) = 21.

Step 3: Add both the result with carry (24 + 21 + 2) = 47

and write down 7 carry 4.

Step 4: Multiply left hand side numbers (3×8) = 24 and

add carry (24 + 4) write down to the left of 7

finally the result we get 2871. (Try to calculate all four

steps in mind.)

Multiplication of a Three digit numbers

(1) 175 x 157 =?

The result of multiplication of three digit number is

175×157 = 27475.

Step 1: Multiply (5×7) = 35 (note down 5 carry 3).

Step 2: Then do cross multiplication (7×7 + 5×5 + 3 (add

carry)) = 77 (note down 7 carry 7).

Step 3: Again (1×7 + 1×5 + 7×5 + 7 (add carry)) = 54 (note

down 4 carry 5).

Step 4: do cross multiplication and add carry (1×5 + 1×7

+ 5 (add carry)) = 17 (note down 7 carry 1).

Step 5: Again (1×1 + 1) = 2, note it down.

And finally the result we get 27475.

(2) 275×354 =?

The result of multiplication of three digit number is

275×354 = 97350.

Step 1: Multiply (4×5) = 20 (note down 0 carry 2).

Step 2: Then do cross multiplication (5×5 + 4×7 + 2 (add

carry)) = 55 (note down 5 carry 5).

Step 3: Again (4×2 + 3×5 + 5×7 + 5 (add carry)) = 63 (note

down 3 carry 6).

Step 4: Again do cross multiplication and add carry (5×2

+ 3×7 + 6) = 37 (note down 7 carry 3).

Step 5: do multiplication of left numbers and add carry

(3×2 + 3) = 9, note it down.

And finally the result we get 97350.

Multiplication of Three and Two digit numbers

(1) 295 x 19 =?

The result of multiplication of three and two digit

number is 295×19 = 5605.

Step 1: Multiply 5×9 = 45 (note down 5 and carry 4).

Step 2: Then do cross multiplication and add carry (9×9

+ 5×1 + 4) = 90 (note down 0 and carry 9).

Step 3: Again do cross multiplication and add carry (2×9

+ 1×9 +9) = 36 (note down 6 and carry 3).

Step 4: Now multiply of left numbers and add carry (2×1

+ 3) = 5, note it down.

And finally the result we get 5605.

(2) 195 x 19 =?

The result of multiplication of three and two digit

number is 195 x 19 = 3705.

Step 1: Multiply 5×9 = 45 (note down 5 and carry 4).

Step 2: Then do cross multiplication and add carry (9×9

+ 5×1 + 4) = 90 (note down 0 and carry 9).

Step 3: Again do cross multiplication and add carry (1×9

+ 1×9 +9) = 27 (note down 7 and carry 2).

Step 4: Now multiply of left numbers and add carry (1×1

+ 2) = 3, note it down.

And finally the result we get 3705.

Multiplication of Four and Two digit numbers

(1)4295 x 19 =?

The result of multiplication of three and two digit

number is 4295×19 = 81605.

Step 1: Multiply 5×9 = 45 (note down 5 and carry 4).

Step 2: Then do cross multiplication and add carry (1×9

+ 9×9 + 4) = 90 (note down 0 and carry 9).

Step 3: Again do cross multiplication and add carry (1×9

+ 9×2 +9) = 36 (note down 6 and carry 3).

Step 4: Again do cross multiplication and add carry (9×4

+ 1×2 +3) =41 (note down 1 and carry 4).

Step 5: Now multiply of left numbers and add carry (1×4

+ 4) = 8, note it down.

And finally the result we get 81605.

(2) 3457 x 23 =?

The result of multiplication of three and two digit

number is 4295×19 = 79511.

Step 1: Multiply 3×7 = 21 (note down 1 and carry 2).

Step 2: Then do cross multiplication and add carry (3×5

+ 2×7 + 2) = 31 (note down 1 and carry 3).

Step 3: Again do cross multiplication and add carry (2×5

+ 3×4 +3) = 25 (note down 5 and carry 2).

Step 4: Again do cross multiplication and add carry (2×4

+ 3×3 +3) = 19 (note down 9 and carry 1).

Step 5: Now multiply of left numbers and add carry (2×3

+ 1) = 7, note it down.

And finally the result we get 79511.