3 Ways to Calculate the Square Root Without the Calculator

2024 Author: Samantha Chapman | [email protected]. Last modified: 2023-12-17 09:13

Calculating the square root of an integer is a very simple operation. There is a logical process that allows you to get the square root of any number even without using the calculator. Before starting, however, it is important to master the basic mathematical operations, that is, addition, multiplication and division.

Steps

Method 1 of 3: Calculate the Square Root of an Integer

Step 1. Calculate the square root of a perfect square using multiplication

The square root of an integer is that number which, multiplied by itself, gives the original starting number as a result. In other words, we can ask ourselves the following question: "What is that number that multiplied by itself gives as a result the root of the square root under consideration?".

• For example, the square root of 1 is equal to 1 precisely because 1 multiplied by itself results in 1 (1 x 1 = 1). Following the same logical reasoning we can say that the square root of 4 is equal to 2 because 2 multiplied by itself gives the result 4 (2 x 2 = 4). Imagine thinking of the square root as a tree; trees grow from their respective seeds and, although they are considerably larger than a seed, they are still closely linked to this small element of nature which is at their root. In the previous example, the number 4 represents the tree while the 2 is the seed.
• Following this logical pattern, the square root of 9 is equal to 3 (3 x 3 = 9), the square root of 16 is 4 (4 x 4 = 16), the square root of 25 is 5 (5 x 5 = 25), the square root of 36 is 6 (6 x 6 = 36), the square root of 49 is 7 (7 x 7 = 49), the square root of 64 is 8 (8 x 8 = 64), the square root of 81 is 9 (9 x 9 = 81) and finally the square root of 100 is 10 (10 x 10 = 100).

Step 2. Use divisions to calculate the square root

To manually calculate the square root of an integer, you can divide it by a series of numbers until you find the divisor that results in itself.

• For example: 16 divided by 4 results in 4. Similarly 4 divided by 2 results in 2 and so on. In these two examples we can say that 4 is the square root of 16 and 2 is the square root of 4.
• Perfect squares result in an integer with no fractional or decimal parts precisely because they derive exclusively from whole numbers.

Step 3. Use the square root symbol

In mathematics, a specific symbol is used to indicate the square root, which is called a radical. It looks like a check mark with a horizontal dash added to the top right.

• N represents the radicand, which is the integer whose square root you want to calculate. The radicand is the argument of the root, so it must be written inside the radical (the root symbol).
• If you have to calculate the square root of 9, you need to start by writing the root symbol (the radical) and inserting the number 9 inside (replacing it with the root "N" of the general formula). At this point, you can draw the equal sign and provide the result, ie 3. The formula in its entirety should be read as follows: "the square root of 9 equals 3".

Method 2 of 3: Calculate the Square Root of Any Positive Number

Step 1. In this case it is necessary to go by trial, discarding the invalid solutions

It is very difficult to calculate the square root of a number that is not a perfect square, but it is still possible.

• Let's assume we need to calculate the square root of 20. We know that 16 is a perfect square whose square root is 4 (4 x 4 = 16). Furthermore, we know that the next perfect square is 25, whose square root is 5 (5 x 5 = 25), so we are sure that the square root of 20 is a number between 4 and 5.
• Let's start by assuming that the square root of 20 is 4, 5. To verify the correctness of our answer we simply have to square 4, 5. In other words we have to multiply it by itself in this way: 4, 5 x 4, 5. At this point, we check if the result is greater or less than 20. If the solution is not the correct one, we will simply have to try another one (for example 4, 6 or 4, 4) until we identify the one that, raised to square, results in precisely 20.
• In our example 4, 5 x 4, 5 = 20, 25, following the logic we must therefore choose a number smaller than 4, 5. Let's try with 4, 4: 4, 4 x 4, 4 = 19, 36. We just found that the square root of 20 is a decimal number between 4, 4 and 4, 5. Let's try to use 4, 445: 4, 445 x 4, 445 = 19, 758. We are getting closer and closer. Continuing to test different numbers following this logical process we will come to find the correct solution that is: 4, 475 x 4, 475 = 20, 03, which we can safely round to 20.

Step 2. Use the average

Also in this calculation process we start by identifying the two perfect squares (one minor and one major) closest to the number whose square root is to be calculated.

• At this point, you must divide the radicand under examination by the square root of one of the two perfect squares identified. Calculate the average between the result obtained and the number used as a divisor (to calculate the average simply add the two numbers under consideration and divide the result by 2). At this point, divide the radicand by the average obtained, then calculate a new average between the previous one and the new result of the division. The number obtained represents the solution to your problem.
• Sounds complex? Perhaps an example will help you understand better. Suppose we want to calculate the square root of 10. The two closest perfect squares to 10 are 9 (3 x 3 = 9) and 16 (4 x 4 = 16). The square roots of these two numbers are respectively 3 and 4. We then proceed by dividing 10 by the square root of the first number, ie 3, obtaining as a result 3, 33. Now we calculate the average between 3 and 3, 33 by adding them together and dividing the result by 2, getting 3, 1667. At this point, we divide 10 by 3, 1667 again; the result is 3.1579. Now let's calculate the average between 3.1579 and 3.1667 by adding them together and dividing the result by 2, we get 3.1623.
• We verify the correctness of our solution (3, 1623) by multiplying it by itself. 3, 1623 x 3, 1623 gives the result 10, 0001, so the solution found is correct.

Method 3 of 3: Calculate the Negative Solution of a Square Root

Step 1. Using the same procedures it is possible to calculate the negative solution of a square root

A square root admits two solutions, one positive and one negative, and we know that multiplying two negative numbers together gives a positive one. Squaring a negative number therefore produces a positive result.

• For example -5 x -5 = 25. It is good to remember that 5 x 5 = 25 too. From this we deduce that the square root of 25 can be either -5 or 5. Basically, the square root of any positive number admits two solutions.
• Similarly 3 x 3 = 9 but also -3 x -3 = 9, so the square root of 9 admits two solutions: 3 and -3. The positive solution is known as the "principal square root", although as we have seen there are two, so, at this point, it is the only result that interests us.

Step 2. Use the calculator

Now that you understand how to manually calculate the square root of a number, you can greatly simplify your life by using a physical calculator or one of the many online applications on the web.

• If you have chosen to use a physical calculator, look for the key marked with the root symbol.
• Online applications will simply ask you to type in the number you want to calculate the square root of and press a button. In a few moments the final solution will appear on the screen without any effort.

Advice

• It might be useful to memorize the series of the first numbers that represent a perfect square:

  • 0² = 0, 1² = 1, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100.
  • If you can, also memorize this sequence: 11² = 121, 12² = 144, 13² 169, 14² = 196, 15² = 225, 16² = 256, 17² = 289.
  • In this case it's easy and fun: 10² = 100, 20² = 400, 30² = 900, 40² = 1600, 50² = 2500.