How to Find Square Root Without Calculator

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Fact-Checked and Reviewed by: Alice Johnson, PhD (Computational Mathematics)

Use the Babylonian Method, one of the oldest and most efficient algorithms, to find the square root of any positive number without relying on modern calculators or the built-in `Math.sqrt()` function.

How to Find Square Root Without Calculator

The Calculated Square Root is:

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How to Find Square Root Without Calculator Formula:

This calculator uses the iterative Babylonian Method (also known as Heron’s Method) to approximate the square root of a number $N$.

$$x_{n+1} = \frac{1}{2} \left( x_n + \frac{N}{x_n} \right)$$

Formula Source: Wikipedia: Methods of Computing Square Roots, Wolfram MathWorld: Heron’s Method

Variables:

  • N (Number): The non-negative number whose square root you wish to find.
  • xn (Current Guess): The current, refined approximation of the square root.
  • xn+1 (Next Guess): The next, more accurate approximation, derived from the current guess.

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What is the Babylonian Method?

The Babylonian method is an ancient technique for finding the square root of a number. It is an iterative process, meaning it generates a sequence of approximations that get closer and closer to the actual square root with each step. It is highly efficient and converges rapidly, which is why it forms the basis for square root calculations in many modern computing systems.

The process starts with an initial arbitrary guess (often $N/2$ or $1$) and then repeatedly calculates a new, better guess by averaging the current guess with the quotient of the number divided by the current guess. This averaging process is key to the method’s accuracy and speed.

How to Calculate Square Root (Example):

Let’s calculate the square root of $N=10$ using the Babylonian Method:

  1. Initial Guess ($\text{x}_0$): Start with an easy guess, let’s say $x_0 = 3$.
  2. Iteration 1: Apply the formula: $x_1 = 0.5 \times (3 + \frac{10}{3}) = 0.5 \times (3 + 3.33333) = 3.16667$.
  3. Iteration 2: Use the new guess: $x_2 = 0.5 \times (3.16667 + \frac{10}{3.16667}) = 0.5 \times (3.16667 + 3.15806) = 3.16237$.
  4. Iteration 3: $x_3$ will be even closer. Continue until the difference between $x_{n+1}$ and $x_n$ is negligible. The true square root of 10 is approximately $3.162277…$

Frequently Asked Questions (FAQ):

Is the Babylonian Method always accurate?
The method provides an increasingly accurate *approximation*. It can get arbitrarily close to the true value, limited only by the number of iterations you perform and the precision of your calculations.

What is a good starting guess?
Any positive number will work, but the closer the initial guess is to the true root, the fewer iterations are required. A common starting point is $N/2$ or simply $1$.

Why does this method work?
The formula always generates a new estimate that is between the current guess ($x_n$) and the quotient ($N/x_n$). One of these values is an overestimate and the other is an underestimate. Averaging them guarantees a better, more accurate value.

Can I find the square root of a negative number?
In real numbers, the square root of a negative number is undefined. This calculator only works for non-negative inputs.

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