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Chapter 2: Whole Numbers

    Introduction

    Whole Numbers

    Representation of Whole Numbers on Number Line

    Properties of Whole Numbers

    Patterns in Whole Numbers

    Exercise 2.1

    Exercise 2.2

    Exercise 2.3

    What Have We Discussed ?

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Chapter 2: Whole Numbers > Exercise 2.3

Exercise 2.3

1. Study the pattern:

1 × 8 + 1 = 9 (1)

12 × 8 + 2 = 98 (2)

123 × 8 + 3 = 987 (3)

1234 × 8 + 4 = 9876 (4)

12345 × 8 + 5 = 98765 (5)

Write the next four steps. Can you find out how the pattern works?

Instructions

Step 6: × + =
Step 7: × + =
Step 8: × + =
Step 9: × + =
Explanation: In each step, the numbers on the left-hand side form digits (1, 12, 123, 1234, etc.). Multiply these numbers by . Add a number on the right-hand side that matches the digit of the number sequence (1, 2, 3, 4, etc.). The result is a sequence of digits: 9, , , , etc.

2. Study the pattern:

91 × 11 × 1 = 1001 (1)

91 × 11 × 2 = 2002 (2)

91 × 11 × 3 = 3003 (3)

Write next seven steps. Check, whether the result is correct. Try the pattern for 143 × 7 × 1, 143 × 7 × 2 and so on.

Instructions

Step 4: 91 × 11 × = , Step 5: 91 × 11 × = ,Step 6: 91 × 11 × =
Step 7: 91 × 11 × = , Step 8: 91 × 11 × = , Step 9: 91 × 11 × = , Step 10: 91 × 11 × =
Checking the Pattern for 143 × 7 × n: 143 × 7 × 1 = while 143 × 7 × 2 = and 143 × 7 × 3 =
Explanation of the Pattern: 91 × 11 = while 143 × 7 =
Both expressions are equivalent to 1001, which forms the basis of the pattern. When you multiply 1001 by different integers (n), the results follow a predictable sequence.

3. How would we multiply the numbers 13680347, 35702369 and 25692359 with 9 mentally? What is the pattern that emerges.

Instructions

For 13680347, find the product: 13680347 × = and then subtract the original number: - 13680347 =
For 35702369, find the product: 35702369 × = and then subtract the original number: - 35702369 =
For 25692359, find the product: 25692359 × = and then subtract the original number: - 25692359 =
Pattern: The resulting numbers are made up of segments, like 123123123, 321321321, and 231231231. This happens because of the structure and digits of the original numbers, which when multiplied by 9, produce cyclic (or) repeating patterns.