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How does sqrt work?

Published in Mathematics 3 mins read

The sqrt function (often denoted as √) calculates the square root of a number, which is a value that, when multiplied by itself, equals the original number.

In simpler terms, the square root operation answers the question: "What number, when multiplied by itself, gives me this number?"

For example:

  • √9 = 3, because 3 * 3 = 9
  • √25 = 5, because 5 * 5 = 25

Here's a breakdown of how it works conceptually:

  • Definition: The square root of a number x is a number y such that y2 = x. Typically, we are interested in the principal square root, which is the non-negative value of y.

  • Mathematical Notation: The radical symbol '√' represents the square root. The number under the radical is called the radicand.

  • Perfect Squares: If the radicand is a perfect square (e.g., 4, 9, 16, 25), the square root is a whole number.

  • Non-Perfect Squares: If the radicand is not a perfect square (e.g., 2, 3, 5, 6), the square root is an irrational number, meaning it cannot be expressed as a simple fraction and has a non-repeating, non-terminating decimal representation. Calculators and computers use algorithms to approximate these square roots.

  • Algorithms for Calculating Square Roots: There are several methods for calculating square roots, including:

    • Newton's Method: This is an iterative method that starts with an initial guess and refines it until it converges to the actual square root.

    • Babylonian Method: A specific case of Newton's method tailored for finding square roots. It iteratively averages a guess with the original number divided by the guess.

    • Digit-by-Digit Method: A method similar to long division that calculates the square root one digit at a time.

  • Implementation in Programming Languages: Most programming languages have a built-in sqrt() function in their math libraries that use highly optimized algorithms to calculate square roots efficiently.

Example using Python:

import math

number = 16
square_root = math.sqrt(number)
print(square_root)  # Output: 4.0

number = 2
square_root = math.sqrt(number)
print(square_root)  # Output: 1.4142135623730951

In summary, the sqrt function effectively reverses the squaring operation, giving you the original number that was multiplied by itself to get the given value. For perfect squares, this is straightforward. For non-perfect squares, algorithms are used to approximate the result.

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