Numbers in general form – Playing with numbers

Numbers in General Form – Playing with  Numbers

Consider the number 45. We write this as

45 = 40 + 5 = (4 x 10) + (5 x 1)

Similarly,

34 = 30 + 4 = (3 x 10) + (4 x 1)

In the way,

354 = 300 + 50 + 4 = (3 x 100) + (5 x 10) + (4 x1)


Now let us consider a 9 digit number, 123456789.  we can write it as,

123456789 = 100000000 + 20000000 + 3000000 + 400000 + 50000 + 6000 + 700 + 80 + 9

= (1 x 100000000) + (2 x 10000000) + (3 x 1000000) + (4 x 100000) + (5 x 10000) + (6 x 1000) + (7 x 100) + (8 x 10) + (9 x 1)

= (1 x 108) + (2 x 107) + (3 x 106) + (4 x 105) + (5 x 104) + (6 x 103) + (7 x 102) + (8 x 101) + (9 x 100)

This is called the base 10 representation of the given numbers or the generalised form of the number.


Consider, for example 136. We can write this in the generalised form as:

136 = (1 x100) + (3 x 10) + (6 x 1)

We see that, 6 is associated with 1; 3 is associated with 10; and 1 is associated with 100. This is the reason, 6 is called the digit in the unit’s place; 3 is the digit in the ten’s place and 1 is the digit in the hundred’s place.


Numbers in general form – Playing with numbers – Exercise 1.1.2

  1. Write the following in the generalised form:

i. 39

Solution:

39 = (3 x 10) + (9 x 1) = (3 x 101) + (9 x 100)


ii. 52

Solution:

52 = (5 x 10) + (2 x 1) = (5 x 101) + (2 x 100)


iii.      106

Solution:

106 = (1 x 100) + (0 x 10) + (6 x 1)

= (1 x 102) + (0 x 101) + (6 x 100)


iv. 359

Solution:

359 = (3 x 100) + (5 x 10) + (9 x 1)

= (3 x 102) + (5 x 101) + (9 x 100)


v. 628

Solution:

628 = (6 x 100) + (2 x 10) + (8 x 1)

= (6 x 102) + (2 x 101) + (8 x 100)


vi. 3458

Solution:

3458 = (3 x 1000) + (4 x 100) + (5 x 10) + (8 x 1)

= (3 x 103) + (4 x 102) + (5 x 101) + (8 x 100)


vii.     9502

Solution:

9502 = (9 x 1000) + (5 x 100) + (0 x 10) + (2 x 1)

= (9 x 103) + (5 x 102) + (0 x 101) + (2 x 100)


viii.    7000

Solution:

7000 = (7 x 1000) + (0 x 100) + (0 x 10) + (0 x 1)

= (7 x 103) + (0 x 102) + (0 x 101) + (0 x 100)


  1. Write the following in the normal form:

i. (5 x 10) + (6 x 1)

Solution:

= 50 + 6 = 56


ii. (7 x 100) + (5 x 10) + (8 x 1)

Solution:

= 700 + 50 + 8

=758


iii. (6 x 1000) + (5 x 10) + (8 x 1)

Solution:

= 6000 + 50 + 8

= 6058


iv. (7 x 1000) + (6 x 1)

Solution:

= 7000 + 6

= 7006


v. (1 x 1000) + (1 x 10)

Solution:

= 1000 + 1

= 1001


  1. Recalling your earlier knowledge, represent 555 in base 5.

Solution:

Numbers in general form - Playing with numbers - Class VIII

[Hint: 4210 are the reminders of 555 when divided by 5]


  1. What is the representation of 1024 in base 2?

Solution:

d

1024 = (1100111000)2

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