Mental Square Root Calculation Techniques

Aug 23, 2024

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Calculating Square Roots in Your Head

Introduction

  • Video by Preshtalwalker on calculating square roots of perfect squares mentally.
  • Historical context: Entrance exam for MIT, 1876.
  • Importance before calculators: Students learned algorithms for mental calculations.

Method Overview

  1. Memorize Squares: Learn the squares of numbers 0-9.
  2. Last Digit Patterns: Identify last digits of squares:
    • 1: 1², 9²
    • 4: 2², 8²
    • 9: 3², 7²
    • 6: 4², 6²
    • 0: 0²
    • 5: 5²

Example Calculations

Example 1: Square root of 1600

  • Last digit: 0 → Write down 0 because 0² = 0.
  • Remaining number: 16; closest square ≤ 16 is 4² = 16.
  • Result: 40.

Example 2: Square root of 4225

  • Last digit: 5 → Write down 5 because 5² = 25.
  • Remaining number: 42; closest square ≤ 42 is 6² = 36.
  • Result: 65.

Example 3: Square root of 841

  • Last digit: 1 (options: 1 or 9). Cross out last two digits → 8.
  • Closest square ≤ 8 is 2² = 4.
  • Square roots to consider: 21² or 29².
    • Check 25² = 625 < 841, so use 29.
  • Result: 29.

Example 4: Square root of 3969

  • Last digit: 9 (options: 3 or 7).
  • Cross out last two digits → 39.
  • Closest square ≤ 39 is 6² = 36.
  • Check 65² → too large, so use 63.
  • Result: 63.

Example 5: Square root of 1156

  • Last digit: 6 (options: 4 or 6).
  • Cross out last two digits → 11.
  • Closest square ≤ 11 is 3² = 9.
  • Check 34² → select smaller option: 34.
  • Result: 34.

Example 6: Square root of 11.56

  • Result: 3.4.

Example 7: Square root of 16384

  • Last digit: 4 (options: 2 or 8).
  • Cross out last two digits → 163.
  • Closest square ≤ 163 is 12² = 144.
  • Check 128: Result: 128.

Extending to Cube Roots

  • Memorize cubes: Similar process as square roots.
  • Example: Cube root of 39304.
    • Last digit: 4 → Write down 4.
    • Remaining: 39; closest cube ≤ 39 is 3³ = 27.
  • Result: 34.

Historical Context

  • Importance of Indian numerals vs. Roman numerals.
  • Calculations with Roman numerals were complex.
  • Adoption of Indian numerals allowed direct calculation.

Conclusion

  • Quote by Laplace on the significance of Indian numerals.
  • Acknowledgment of community support on YouTube.
  • Encouragement to continue learning and solving problems.