Square Root Calculator

This calculator can be used to find the square root of the input number. Input numbers can be positive or negative, and the root calculator will identify the principal square root of the number and the opposite root.

Directions for use

To use the square root calculator, enter the given number, and press “Calculate.” The calculator will return the principal square root of the number and the opposite (negative) square root. It will also indicate whether the input number is a perfect square.

Squares and square roots

Squares

A square of a given number is the number multiplied by itself. For example, 3 × 3 = 9, which means that the square of 3 is 9, or three squared equals 9. The square of a number is usually written as follows: x². So, if x = 3, the previous equation can be written as 3² = 9. Some examples of squares of different numbers are presented below:

Number	Square
2	4
3	9
4	16
5	25
0.1	0.01
12	144

Look at negative numbers and find (-3)². (-3)² = (-3) × (-3) = 9, since multiplying two negative signs returns the positive sign. Therefore, (-3)² = 3² = 9.

Perfect squares

A perfect square is a square of an integer; for example, 4, 9, 16, and 25 are all perfect squares. Below are the perfect squares of the first integers. It is useful to remember them.

Number	Square
0	0
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144

Consequently, if the square root of a number is an integer, that number is a perfect square. The calculator on this page will indicate whether the input number is a perfect square.

Square roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square roots of 9 are 3 and -3, since 3 × 3 = 9, and (-3) × (-3) = 9, therefore, (-3)² = 3² = 9. Similarly, the square roots of 16 are 4 and -4, and so on. Each number (except 0) has two square roots – positive and negative square roots.

The positive square root of a number is called the principal square root; when it is not specified which square root has to be calculated, the principal root is usually implied. For example, in the question “What is the square root of 36?”, we are only asked to find ONE square root, so we only take the principal root into account, and the answer will be “6”.

The square root symbol

The square root symbol is called the radical and is depicted as follows: √. So, to write the square root of 16 mathematically, we will write √16.

According to a strict mathematical definition, for any function f(x, y), there must be a unique value of y for each value of x. Imagine we have a function where y equals the square root of x. Then, for each value of x, there would be two values of y – a positive square root and a negative square root. This is against the mathematical definition of a function! To work around this problem, mathematicians have assigned the radical symbol √ only to the principal root.

This means that while the square roots of 16 are 4 and -4, mathematically, √16 = 4. This has to be taken into account when solving mathematical equations. Any equation of the type y² = x has two solutions, written as y = √x and y = -√x, or y = ±√x.

Square roots of negative numbers

In the section above, we have demonstrated that the square of any real number is always positive. If the number is positive, its square is also positive. And if a number is negative, its square is still positive since multiplying two negative signs returns a positive sign.

Now let’s imagine that there is a number that gives a negative result when squared. Numbers that give negative results when squared are called imaginary numbers. The basic imaginary number is i, defined as:

i² = -1

or

i = √(-1)

Let’s try to find the square roots of (-4):

√(-4) = √(4 × (-1)) = √4 × √(-1) =2 × i = 2i

The principal square root of (-4) is 2i. And if we take the opposite square root of 4 (-√4 = -2) into account in the above equation, we will also get the opposite solution: -2i.

How to find the square root of a number

Calculating square roots of perfect squares is relatively easy. But calculating square roots of decimals, or integers that are not perfect squares, can be tricky. Several ways of calculating square roots, including a calculation method allowing one to find the exact square root of any number, are explained on this page.

Real-life application

John is planning to rent a studio apartment. He has found an advertisement for a studio with an area of 20.25 square meters. How can he estimate the length of the studio walls to visualize the size of the place better?

Solution

In real estate, the sizes of apartments, houses, and land areas are usually given in square meters. At the same time, some listings include the corresponding lengths, but many don’t. It can be hard to visualize the size of the space by considering the square meters of area. But, if we imagine the total area as a square with a side of a certain length, we will have a better idea of how big the place is. To do that, we have to extract the square root of the total area:

√20.25 = 4.5

Note that we are talking about the physical size of an apartment. Therefore, we will only need the principal square root.

It is also interesting to note that extracting square roots works with dimensions! This example measured the total area in square meters (m²). When we are finding the length of a wall, we are technically taking a square root of 20.25 m²:

√(20.25 m²) = √20.25 √(m²) = 4.5 m

Answer

A studio with an area of 20.25 square meters can be visualized as a square room, with each wall 4.5 meters long.