ICSE Class 8 Maths Numbers Chapter 01 Number Systems

Number Systems

You are already familiar with numbers. You have read about natural numbers, whole numbers and integers, and the four fundamental operations on them. Let us briefly revise what you have already learnt.

The numbers used to count things are called natural numbers. The set of natural numbers N = {1, 2, 3, 4, ...}.

The number 0 (zero) together with the natural numbers gives us the set of whole numbers W = {0, 1, 2, 3, 4, ...}.

The members of the collection of numbers ..., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ... are called integers. 1, 2, 3, 4, ..., are called positive integers while -1, -2, -3, -4, ..., are called negative integers. The integer 0 (zero) is neither positive nor negative. The set of integers is denoted by I or Z. So, Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}. Integers are also called signed numbers or directed numbers.

Four Fundamental Operations on Directed Numbers

The directed numbers and the integers are the same and you are already familiar with the four fundamental operations on integers.

Simplification

An expression involving integers (i.e., directed numbers) is simplified by the rule of BODMAS.

Steps 1. First, carry out the operations inside the brackets (B). Brackets are removed in the order: line bracket (or vinculum), first (or common) brackets, second (or curly) brackets, third (or rectangular or square) brackets.

2. Then change 'of' into 'x' and multiply (O).

3. Carry out the operations of division and multiplication, in that order (DM).

4. Finally, carry out the operations of addition and subtraction (AS).

Example: Simplify 14 + 16 of [625 - (55 - 5 x (9 - 6 - 3))].

Solution: The given expression = 14 + 16 of [625 - (55 - 5 x (9 - 3))]

= 14 + 16 of [625 - (55 - 5 x 6)] = 14 + 16 of [625 - (55 - 30)]

= 14 + 16 of [625 - 25] = 14 + 16 of 25 = 14 + 400 = 414.

Teacher's Note

Understanding order of operations (BODMAS) helps students solve multi-step problems correctly, just like following a recipe ensures the right cooking outcome.

Solved Examples

Example 1

Write all possible three-digit numbers using the digits 0, 3 and 5 when the digits are (i) not repeated, (ii) repeated.

Solution: The digit 0 cannot be placed in the leftmost place as it would yield a two-digit number.

(i) When the digits are not repeated

(ii) When the digits are repeated

Teacher's Note

This exercise demonstrates how the position of digits changes the value of numbers, similar to how the arrangement of items in a store window creates different displays.

Example 2

Find the smallest five-digit number which is exactly divisible by 241.

Solution: The smallest five-digit number = 10000.

241) 10000 (41

- 964

360

- 241

119 10000 divided by 241 gives the remainder 119.

So, the required number must be greater than 10000 by (241 - 119). Hence, the required number = 10000 + (241 - 119) = 10000 + 122 = 10122.

Teacher's Note

Finding divisible numbers is used in real life, such as distributing items equally into groups without remainder.

Example 3

Find the largest six-digit number which is exactly divisible by 459.

Solution: The largest six-digit number = 999999.

459) 999999 (2178

- 918

819

- 459

3609

- 3213

3969

- 3672

297 999999 divided by 459 gives the remainder 297.

Hence, the required number = 999999 - remainder = 999999 - 297 = 999702.

Teacher's Note

This method helps find the exact capacity of containers or spaces when we need the largest amount that fits perfectly.

Example 4

Simplify 4 of [16 - (72 - (-3 + 15 x 3)) + 2].

Solution: The given expression = 4 of [16 - (72 - (12 x 3)) + 2]

= 4 of [16 - (72 - 36) + 2] = 4 of [16 - 2 + 2]

= 4 of 16 = 4 x 16 = 64.

Teacher's Note

Careful bracket simplification prevents errors in complex calculations, just as following steps carefully in a scientific experiment ensures accurate results.

Example 5

Simplify 24 - [36 - 9 - 4 of (11 - 5 - 13)].

Solution: The given expression = 24 - [36 - 9 - 4 of {11 - (-8)}] = 24 - [36 - 9 - 4 of {11 + 8}]

= 24 - [36 - 9 - 76] = 24 - [4 - 76]

= 24 - [-72] = 24 + 72 = 96.

Teacher's Note

Understanding negative numbers and their operations helps in contexts like temperature changes and bank account balances.

Example 6

Fill in the blanks by selecting from the brackets.

(i) If a profit of Rs 100 is described as + 100 then a loss of Rs 80 will be described as ....... . [-20, +80, -80] (We use Rs to represent Rs or Re.)

(ii) -15 - (-18) ....... . (-2) x 5 [=, <, >]

(iii) -7 x (-4) + 56 - (-2) = ...... . [-56, 0, 56]

Solution:

(i) -80

(ii) > {-: -15 - (-18) = -15 + 18 = 3; (-2) x 5 = -10; and 3 > -10}

(iii) 0 {-: -7 x (-4) + 56 - (-2) = 28 + (-28) = 0}

Teacher's Note

Real-world applications of directed numbers like profit-loss and temperature changes help students relate abstract concepts to everyday situations.

Exercise 1

1. Find (i) the sum and (ii) the difference of the smallest whole number and the smallest natural number.

2. Write all the possible 4-digit numbers using each of the digits 4, 0, 5, 9 only once.

3. Write all the possible 3-digit numbers using each of the digits 2, 5, 9 only once.

4. Write all the possible 2-digit numbers using digits from 2, 0, 5 when the digits (i) may be repeated, (ii) are not repeated.

5. Find the sum of the smallest and the greatest numbers of 6 digits using each of the digits 0, 1, 2, 7, 5, 9 only once.

6. Find the smallest six-digit number which is exactly divisible by 471.

7. Find the largest seven-digit number which is exactly divisible by 313.

8. Find the largest five-digit number which is exactly divisible by 37.

9. Fill in the blanks by using = or -.

(i) 100 - 8 x 24 + 3 ....... 100 - (8 x 24) + 3 (ii) 4 x 5 - 3 + 12 - 6 ...... 4 x 5 - 3 + 12 - 6

10. Use > or < to fill in the blanks.

(i) 2 + 5 x 8 + (-24) + 4 ...... 2 + 5 x 8 + (-24) + 4 (ii) 2 x 3 + 9 - 6 + 3 ...... 2 x 3 + 9 - 6 + 3

11. Simplify the following.

(i) 260 - 5 of [50 - {16 + (24 - 4 - 1)}] (ii) 350 - [64 - 8 - 7 of {1 - 6 - 2 - (9 - 1 - 3)}]

12. Simplify -7 - [(-8) - 6 + (-2) - {10 of (-2) - (7 - 17)}].

Answers

1. (i) 1 (iii) -1

2. 4590, 4509, 4950, 4905, 4059, 4095, 5490, 5409, 5940, 5904, 5049, 5094, 9450, 9405, 9540, 9504, 9045, 9054

3. 259, 295, 529, 592, 925, 952 4. (i) 20, 22, 25, 50, 52, 55 (ii) 20, 25, 50, 52

5. 1077789 6. 100323 7. 9999724 8. 99974

9. (i) = (ii) - 10. (i) > (ii) > 11. (i) 2 (ii) 349 12. -12

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