Square Root Calculator

The square root of a number is the value that when multiplied by itself equals the original number. When you enter a number into this calculator, it will automatically find both the principal (the positive) and negative square roots, tell you whether or not the number is a perfect square, and convert the result to a simplified radical form.

Squaring and square roots are inverse operations of each other. If you square 5, you get 25, so the square root of 25 is 5. The problem with this is that most numbers don't have whole number square roots. In those cases, estimates, simplifications or even complex numbers are needed.

What is a square root?

Given a number x, the square root of x is a number y such that y squared equals x. Since the product of two negative numbers yields a positive number, both the squares of positive and negative values will result in the same outcome. Therefore, every positive number has a square root.

Let's take 9 as an example. Both +3 and -3 squared give 9, therefore the square roots are +3 and -3. The positive one is called the principal square root, and when the radical sign stands alone it represents the principal square root. Symbolically it looks like this:

$\sqrt{x} = y \iff y^2 = x$

Zero is an exception. The square root of zero is only itself. There's also a practical way to write the square root as a power of 0.5. For this reason, the following two forms have exactly the same meaning.

$\sqrt{x} = x^{1/2} = x^{0.5}$

How to manually calculate a square root:

The fastest method is to recognize if a number is a perfect square, that is, one whose square root is an integer. It helps to memorize these numbers.

If a number is not a perfect square, then its square root will lie between two consecutive whole numbers. For example, the square root of 52 lies between the square roots of 49 (which is 7) and 64 (which is 8). Since 52 is closer to 49 than it is to 64, we can reasonably estimate that its square root will be close to 7.2.

To further improve the estimate, one can use the Babylonian method of averaging. One divides the number by its estimated square root and takes the average of that estimate and the quotient. This process is repeated. With each iteration, approximately twice as many digits are correct.

If you calculate the square root of 27, you can see how quickly it converges. You start with an estimate of 5.125. You divide 27 by 5.125 to get 5.268. The average of these two numbers is 5.197. Another iteration gives 5.196, and at this point the value doesn't change much. So the square root of 27 is approximately 5.196.

Another trick is to simplify first and use known square roots of prime numbers. These square roots are usually rounded to two decimal places.

52 can be expressed in the form by multiplying 2 with the square root of 13, and the square root of 13 is replaced by approximately 3.61. When they are multiplied together, it becomes approximately 7.22, which matches the previous estimated result.

Simplify Square Roots:

The same value can be expressed in different radical terms but the simplest form minimizes the number inside the radical as much as possible. This form is a product of an integer and a small radical which is written as "a times the square root of b".

This is possible because the square root of a product equals the product of the square roots of the factors.

The square root of a product is equal to the product of the square roots of each factor. $\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}$

Factor a number into the product of a perfect square and another number, then take the square root of the perfect square. For example, 8 can be factored as 4 times 2. Since the square root of 4 is 2, the square root of 8 is equal to 2 times the square root of 2.

This square root can be simplified as follows: $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4}\,\sqrt{2} = 2\sqrt{2}$

Let's look at two more examples. Since 27 is the product of 9 and 3, its square root will be the product of 3 and the square root of 3. On the other hand, since 72 contains the factor 36, its square root will be the product of 6 and the square root of 2. This corresponds exactly to the decompositions shown in the figure above.

Simplification is only possible if a number contains perfect square factors. The square root of 15 cannot be simplified, because 15 does not have any perfect square factors other than 1. The factors of 15 are 1, 3, 5 and 15, but none of these factors are perfect squares. The table at the top of this article will break down numbers by their factor pairs, automatically extracting the perfect square factors.

What is a perfect square?

A perfect square is a whole number whose square root is also a whole number. For example, 4, 9, 16 and 25 are the squares of 2, 3, 4 and 5 respectively. If your inputted number happens to be one of these, this calculator will tell you so.

If you know your perfect squares by heart, you can quickly recognize whether the square root of a number is simple or complex. If a number falls between two perfect squares, its square root will be an irrational number that never ends. This is true for the square roots of 2 and 5.

Square roots of negative numbers

Negative numbers do not have a square root in the set of real numbers because there is no real number that squares to give a negative number. Mathematicians introduced the imaginary unit i to solve this problem. i is defined as the number whose square is −1.

$i = \sqrt{-1}$

Using "i", the square root of any negative number can be calculated as the product of the square root of its absolute value and "i". So, for example, the square root of -9 is 3i and the square root of -49 is 7i. If a negative number is entered, this calculator will automatically switch to this complex number format.

Addition, multiplication and division of roots

Adding roots is different from regular addition. If you add the square root of 2 and the square root of 3, it's not the same thing as the square root of 5. Only numbers under the same radical can be combined, and in this case the coefficients are added together. So the sum of 3 times the square root of 2 plus 5 times the square root of 2 is equal to 8 times the square root of 2.

Multiplication and division are easier because you can just multiply or divide the numbers under the radical signs.

$\sqrt{x}\,\sqrt{y} = \sqrt{x \cdot y} \qquad \frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\tfrac{x}{y}}$

For example the square root of 3 times the square root of 2 is equal to the square root of 6, and the square root of 15 divided by the square root of 3 equals the square root of 5.

Is the square root of a rational number or an irrational number?

The square root of a perfect square is an integer and therefore rational. All other square roots are irrational. This means they cannot be written as the ratio of two integers and have no finite decimal representation or repeating pattern.

The square root of 2 is a typical example that was first determined on Babylonian clay tablets as 1.41421. For the same reason, the square roots of 3, 5 and 7 are also irrational. A quick method to determine this is that the square root will only be rational if the whole number can be written as a power of an even number. In other words: The number must be a perfect square.

A short history of roots.

The square root was studied by the Babylonians, Egyptians, Indians, Greeks and Chinese long before the symbol for it existed. It is believed that the radical sign we use today comes from the lowercase letter r, which represents the Latin word radix meaning "root".

The horizontal line over the numbers inside the radical sign is called a vinculum and means "connection" in Latin. So, the word "radical" has nothing to do with "extreme". It refers to the root or basis of a number. The term "square" comes from a geometric concept. The square root of x is the length of one side of a square whose area is x.