Basic Math Class (Number Part-III)

Numbers-III

VI.  MULTIPLICATION BY SHORT CUT METHODS

1.  Multiplication By Distributive Law :

        (i) a x (b + c) = a x b + a x c

       (ii) a x (b-c) = a x b-a x c.

   Ex.   

       (i)    567958 x 99999 

           = 567958 x (100000 - 1)

           = 567958 x 100000 - 567958 x 1 

           = (56795800000 - 567958) 

           = 56795232042. 

      (ii)     978 x 184 + 978 x 816 

             = 978 x (184 + 816) 

             = 978 x 1000 

             = 978000.

2.  Multiplication of a Number By 5ⁿ:  

Put n zeros to the right of the multiplicand and divide the number so formed by 2ⁿ

       Ex. 975436 x 625 

          = 975436 x 5⁴

          = 9754360000 

          = 609647600

VII.   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. 

VIII.  DIVISION ALGORITHM OR EUCLIDEAN ALGORITHM

If we divide a given number by another number, then :

      Dividend = (Divisor x Quotient) + Remainder

IX.    (i) (xⁿ - aⁿ ) is divisible by (x - a) for all values of n.

            (ii) (xⁿ - aⁿ) is divisible by (x + a) for all even values of n.

           (iii) (xⁿ + aⁿ) is divisible by (x + a) for all odd values of n.

X. PROGRESSION

A succession of numbers formed and arranged in a definite order according to certain definite rule, is called a progression.

1. Arithmetic Progression (A.P.) : 

If each term of a progression differs from its preceding term by a constant, then such a progression is called an arithmetical progression. This constant difference is called the common difference of the A.P. An A.P. with first term a and common difference d is given by a, (a + d), (a + 2d),(a + 3d),.....

The nth term of this A.P. is given by T_n =a (n - 1) d.

The sum of n terms of this A.P.

S_n = n/2 [2a + (n - 1) d] = n/2   (first term + last term).

SOME IMPORTANT RESULTS :

 (i)   (1 + 2 + 3 +…. + n) =n(n+1)/2

(ii)   (l² + 2² + 3² + ... + n²) = n (n+1)(2n+1)/6

(iii)  (1³ + 2³ + 3³ + ... + n³) =n²(n+1)²

2. Geometrical Progression (G.P.): 

A progression of numbers in which every term bears a constant ratio with its preceding term, is called a geometrical progression. The constant ratio is called the common ratio of the G.P. A G.P. with first term a and common ratio r is :

a, ar, ar²,

In this G.P. T_n = ar^n-1

sum of the n terms, Sn=   a(1-rⁿ)

                                          (1-r)

To be Continued........