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# Square Root Calculator

Square root is the reverse operation of squaring the number i.e. for square the number we use $$x^2$$ and for square root the number we use $$x^{1/2}$$. Example: $$\sqrt{36} = 6$$ (because  $$6 × 6 = 36$$, so the square root of 36 is 6. Note that (−6) × (-6) = 36 too, so -9 is also a square root of 36. The radical (i.e. √) which always means the positive square root.

$$\sqrt{x} = ?$$

## What is a square root symbol called?

The square root symbol $$\sqrt 2$$ is called the ‘Radical’

The number that is written inside the radical symbol is called ‘Radicand’.

### Square Root Symbol:

$$\sqrt N$$ is commonly represented by the square root of a number N.
It can also be represented by:   $$2 \sqrt N$$
In exponential notation, it can be represented by: $$N \frac {1}{2}$$
“Square root” can be abbreviated as “sqrt”.

## How to calculate the square of :

• ### Positive numbers:

The square root of the positive numbers can be calculated by:
We take the positive number 100.

Put it in square root $$\sqrt 100$$

We need a value which is multiplied by itself to give the original number and that is 10, like

$$10 \times 10 = 10^2 = 100$$ so, the $$10$$ is a perfect square of $$100$$

• ### A Negative Number

Multiply by “i” after finding the square root of the same positive number. ( where i represents an imaginary number and $$i = square root of -1)$$ For calculating the square root of a negative number, Example: square root of -7.

$$= (square root of 7) × (square root of -1)$$

$$= (square root of 7) × (i)$$

$$= 2.64575131 × i$$

$$= 2.64575131i$$

It can also be calculated by above calculator.

• ### Imaginary number:

A complex number is an Imaginary number it can be written as a real number multiplied by the imaginary unit i, which is defined by its property $$i2 = −1$$. The square of an imaginary number bi is $$−b2$$. For example, $$3i$$ is an imaginary number, and its square is $$−9$$. (0) zero is examined to be both real and imaginary.

• ### Calculation of an imaginary number:

For calculating the square root of an imaginary number, find the square root of the number as if it were a real number (without the i) and then multiply by the square root of i (where the square root of $$i = 0.7071068 + 0.7071068i$$)

Example: Square root of 5i

$$= (square root of 5) × (square root of i)$$

$$= (2.236068) × (0.7071068 + 0.7071068i)$$

$$= 1.5811388 + 1.5811388i$$

## Square Root Calculator:

This calculator helps to find the principal square root and roots of real numbers. Inputs for the radicand x can be positive or negative real numbers. The answer will also tell you if you entered a perfect square.

## Positive and negative square root:

Any positive real number has two square roots, one positive and one negative. For example, the square roots of 4 are -2 and +2, since $$(-2)2 = (+2)2 = 4$$.

## Principal square root:

Any non-negative real number x has a unique nonnegative square root r; this is called the principal square root. For instance, the principal square root of 4 is $$+\sqrt{4} = +2$$, while the other square root of 4 is $$-\sqrt{4} = -2$$. In usual operation, unless otherwise specified, “the” square root is habitually taken to mean the principal square root.”.

## Some basic Perfect squares are:

$$\sqrt 1 = 1^2 \pm 1$$

$$\sqrt 4 = 4^2 \pm 2$$

$$\sqrt 9 = 3^2 \pm 3$$

$$\sqrt 16 = 4^2 \pm 4$$

$$\sqrt 25 = 5^2 \pm 5$$

$$\sqrt 36 = 6^2 \pm 6$$

$$\sqrt 49 = 7^2 \pm 7$$

$$\sqrt 64 = 8^2 \pm 8$$

$$\sqrt 81 = 9^2 \pm 9$$

$$\sqrt 100 = 10^2 \pm 10$$

## Some basic Non-perfect squares are:

Members of the irrational numbers are those that are not a perfect square of the square roots of numbers. We cannot write it as the quotient of two integers. The decimal form of an irrational number will neither conclude nor repeat. The irrational numbers with one accord with the rational numbers compose the real numbers.

$$\sqrt 2 = \pm 1.41421356$$

$$\sqrt 3 = \pm 1.73205081$$

$$\sqrt 5 = \pm 2.23606798$$

$$\sqrt 6 = \pm 2.44948974$$

$$\sqrt 7 = \pm 2.64575131$$

$$\sqrt 8 = \pm 1.41421356$$

$$\sqrt 10 = \pm 3.16227766$$

$$\sqrt 11 = \pm 3.31662479$$

## Uses of Square Root:

• It can be used to solve  the distance between two points (Pythagorean Theorem).
• It can be used for the length of a side of a right triangle (Pythagorean Theorem).
• It can be used to find solutions to quadratic equations.
• It can be used to find normal distribution.
• It can be used to find the standard deviation.
• Basically to solve for a squared variable in an equation

## Why are square roots necessary?

It defines major concept of standard deviation used in probability theory and statistics. It has a central use in the formula of roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots.It is necessary for algebra and geometry.

## How square root makes your life simple:

Suppose you need to buy a new apartment, obviously, you will have the idea on the size of it. Now, you see an advertisement saying 900 square foot apartment, how will you imagine it?

You use the simple “square roots” 900 square foot- 30 * 30 foot! That’s how square roots make your life simple!! This is one of the uncomplicated condition and you can observe a lot more in your habitual life.

## Diagonal measurement by Pythagorean theorem:

To find the diagonal measurement of a square apartment the measurements of the length and the width must be known:

The measurements will be found on the foundation plan or the floor plan. Finding the square root of the length2 + width2
will give the diagonal dimension of the apartment.

Finding the square root of the length2 + width2 will give the diagonal dimension of the apartment.

Example:

Foundation of an apartment by Pythagorean theorem:

$$a^2 + b^2 = c^2$$
$$=> (19)^2 + (14)^2 = c^2$$
$$=> 341 + 196 = c^2$$
$$=> c \sqrt 537$$
$$c = 23.1732605ft$$

## Estimation Method to find Square Root by Carpenters:

When carpenters use the Pythagorean Theorem to square an apartment.it is important that the diagonal measurement be correct to the nearest $$\dfrac{ 1 }{ 16 }th$$ of an inch. Carpenters cannot estimate the diagonal because the house will not be square.

Converting 23.1732605 ft. to the nearest $$\dfrac{ 1 }{ 16 }th$$ inch:

Take off the whole number 23 ft.

Multiply the decimal by 12 to get inches $$0.173260512 = 2.079126in$$.

Take off the whole number 2in.

Multiply the decimal by 16 to get the nearest $$\dfrac{ 1 }{ 16 }$$

$$0.079126 16 =1.266016$$ in.

23.1732605ft. OR 23ft. OR $$\dfrac{ 1 }{ 16 }$$ inch.

## Square Root by Nearest Estimation Method:

Estimate the square root of 5.

1. Pick two perfect squares closest to the number you want to find the square root of.
2. Choose one perfect square which is greater ‘<’ than the number you want to find the square root of and one perfect Square which is ’>’ less than the number you want to find the square root of.
3. Two perfect squares below and above 5 are 4 and 9.
4. Since 5 is closer to 4 than it is to 9, then 5 must be between $$=> \sqrt 4$$ = 2 and $$=> \sqrt 9$$ = 3 but closer to $$=> \sqrt 4$$ = 2
5. An estimate around 2.4 to 2.5 would b fine.
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