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 10^{8}) + (2 x 10^{7}) + (3 x 10^{6}) + (4 x 10^{5}) + (5 x 10^{4}) + (6 x 10^{3}) + (7 x 10^{2}) + (8 x 10^{1}) + (9 x 10^{0})

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.

**Exercise 1.1.2**

- Write the following in the generalised form:

i. 39

**Solution:**

39 = (3 x 10) + (9 x 1) = (3 x 10^{1}) + (9 x 10^{0})

ii. 52

**Solution:**

52 = (5 x 10) + (2 x 1) = (5 x 10^{1}) + (2 x 10^{0})

iii. 106

**Solution:**

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

= (1 x 10^{2}) + (0 x 10^{1}) + (6 x 10^{0})

iv. 359

**Solution:**

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

= (3 x 10^{2}) + (5 x 10^{1}) + (9 x 10^{0})

v. 628

**Solution:**

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

= (6 x 10^{2}) + (2 x 10^{1}) + (8 x 10^{0})

vi. 3458

**Solution:**

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

= (3 x 10^{3}) + (4 x 10^{2}) + (5 x 10^{1}) + (8 x 10^{0})

vii. 9502

**Solution:**

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

= (9 x 10^{3}) + (5 x 10^{2}) + (0 x 10^{1}) + (2 x 10^{0})

viii. 7000

**Solution:**

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

= (7 x 10^{3}) + (0 x 10^{2}) + (0 x 10^{1}) + (0 x 10^{0})

- 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

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

**Solution:**

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

- What is the representation of 1024 in base 2?

**Solution:**

1024 = (1100111000)_{2}

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