Finding square root through repeated subtraction

5.5.2 Finding square root through repeated subtraction

Description

Quick Overview

The section explains how to find square roots using the method of repeated subtraction of odd natural numbers.

Standard

In this section, students learn that square roots can be found by subtracting consecutive odd natural numbers from a perfect square until reaching zero. This method demonstrates the relationship between square numbers and odd numbers.

Detailed

Finding Square Roots through Repeated Subtraction

In this section, we explore a unique method to find the square root of a perfect square by utilizing the principle that the sum of the first n odd natural numbers equals nĀ². By starting with a perfect square and subtracting successive odd numbers, we can ultimately derive the square root.

Key Concepts:

  1. Repeated Subtraction: The process involves subtracting 1, 3, 5, 7, and so forth from the perfect square. The total number of subtractions made until the remaining value reaches zero gives the square root.
  2. Example: Taking the number 81 as an example:
  3. 81 - 1 = 80
  4. 80 - 3 = 77
  5. 77 - 5 = 72
  6. 72 - 7 = 65
  7. 65 - 9 = 56
  8. 56 - 11 = 45
  9. 45 - 13 = 32
  10. 32 - 15 = 17
  11. 17 - 17 = 0

After 9 subtractions, we reach 0, thus confirming that the square root of 81 is 9.
3. The process of finding square roots through this method highlights the deeper relationship between square numbers and odd counting numbers, showcasing a remarkable property in mathematics.

Key Concepts

  • Repeated Subtraction: The process involves subtracting 1, 3, 5, 7, and so forth from the perfect square. The total number of subtractions made until the remaining value reaches zero gives the square root.

  • Example: Taking the number 81 as an example:

  • 81 - 1 = 80

  • 80 - 3 = 77

  • 77 - 5 = 72

  • 72 - 7 = 65

  • 65 - 9 = 56

  • 56 - 11 = 45

  • 45 - 13 = 32

  • 32 - 15 = 17

  • 17 - 17 = 0

  • After 9 subtractions, we reach 0, thus confirming that the square root of 81 is 9.

  • The process of finding square roots through this method highlights the deeper relationship between square numbers and odd counting numbers, showcasing a remarkable property in mathematics.

Memory Aids

šŸŽµ Rhymes Time

  • Subtract the odds, one by one, until you reach zeros, you've just begun!

šŸ“– Fascinating Stories

  • Once, a brave knight tried to find the roots of his kingdom's perfect square. Each odd number he conquered brought him closer to zero, leading him back to nine.

šŸ§  Other Memory Gems

  • R.O.O.T.: Repeatedly Oddly Overcome to Zero.

šŸŽÆ Super Acronyms

R.S.R. - Repeated Subtraction Reaches zero.

Examples

  • Example 1: Finding the square root of 81 by subtracting odd numbers and reaching 0 after 9 subtractions.

  • Example 2: If you start with 36 and use repeated subtraction: 36 - 1, 36 - 3, and so on, you will find that you will reach 0 after 6 steps.

Glossary of Terms

  • Term: Square Root

    Definition:

    A value that, when multiplied by itself, gives the original number.

  • Term: Repeated Subtraction

    Definition:

    A method of finding square roots by continuously subtracting odd numbers from a perfect square until zero is reached.

  • Term: Odd Natural Numbers

    Definition:

    The sequence of numbers that are not divisible by 2, e.g., 1, 3, 5, 7, etc.