How to calculate the square root without a calculator

Imagine you find yourself in a situation where you need to the square root of a number, and you don’t have access to a . Although it may seem daunting at first, there are methods to determine the square root paper and pen. In this article, we will explore two fundamental techniques: the long division method and the approximation method.

1. Long Division Method:
The long division method is a step-by-step process to calculate the square root manually. Let’s go through the process with an example:

Step 1: Start by separating the digits of the number into pairs, starting from the decimal point if there is one. For instance, if we want to find the square root of 537, it becomes 05, 37.

Step 2: Find the largest digit ‘x’ such that when added to the current partial root (initially 0), the result ‘r’ is less than or equal to the first pair. In this case, the largest digit is 2, as 20 squared is 400, which is less than 537.

Step 3: Subtract the square of ‘x’ from the first pair and bring down the next pair to the right. The result is our new dividend.

Step 4: Double the current partial root and append any digit ‘y’ on the right. Multiply the new root by ‘y’ and find the largest ‘y’ such that the product is less than or equal to the new dividend. In this case, the new dividend is 137 and, assuming our partial root is 20, we find that the largest digit ‘y’ is 6, as 266 squared is 70916, which is less than 13700.

Step 5: Repeat steps 3 and 4 until all pairs have been exhausted or your desired level of precision is reached.

By following the steps, we can determine the square root of 537 to be approximately 23.19.

2. Approximation Method (Babylonian Method):
The approximation method is iterative and works by refining an initial guess repeatedly until finding the desired precision. It follows these steps:

Step 1: Start by guessing an initial value for the square root. For example, if we want to find the square root of 537, let’s start with an initial guess of 10.

Step 2: Divide the number by the initial guess obtained in Step 1.

Step 3: Get the average of the result in Step 2 and the initial guess, and use it as a new approximation.

Step 4: Repeat steps 2 and 3 several times until reaching the desired level of accuracy.

By applying the approximation method, the square root of 537 is approximately 23.166.

Remember, both methods provide approximations of the square root. Calculating square roots precisely can be extremely time-consuming and challenging without a calculator. It is crucial to understand that if high precision is necessary, a calculator or specialized software is the most reliable tool to achieve accurate results.

In conclusion, calculating the square root without a calculator is possible with the help of the long division method and the approximation method. By following the step-by-step procedures outlined in this article, you can obtain rough approximations of square roots. However, for precise calculations, it is best to rely on calculators or software specifically designed for such calculations.