Square Root Calculator

Calculate the square root of any non-negative number.

Enter a number in the field below to find its square root.

Examples

Here are some examples of how to use the calculator.

Perfect Square

perfectSquare

Calculate the square root of a perfect square.

Number: 16

Non-Perfect Square

nonPerfectSquare

Calculate the square root of a non-perfect square.

Number: 2

Zero

zero

Calculate the square root of zero.

Number: 0

Large Number

largeNumber

Calculate the square root of a large number.

Number: 123456789

Other Titles
Understanding Square Roots: A Comprehensive Guide
An in-depth look at square roots, their applications, and how to calculate them.

What is a Square Root?

  • Definition of a Square Root
  • Perfect Squares
  • Principal Square Root
A square root of a number 'x' is a number 'y' such that y² = x. In other words, a number y whose square (the result of multiplying the number by itself, or y × y) is x. For example, 4 and -4 are square roots of 16 because 4² = (-4)² = 16. Every non-negative real number x has a unique non-negative square root, called the principal square root, which is denoted by √x, where √ is called the radical sign or radix.
Perfect Squares
A perfect square is an integer that is the square of an integer. For example, 25 is a perfect square because it is the product of 5 multiplied by itself (5 × 5). The square root of a perfect square is always an integer.
Principal Square Root
For any positive number, there are two square roots: one positive and one negative. The principal square root is the positive one. When we write √x, we are referring to the principal square root. For example, √16 = 4.

Examples of Square Roots

  • √25 = 5
  • √144 = 12
  • √0 = 0

Step-by-Step Guide to Using the Square Root Calculator

  • Inputting Your Number
  • Calculating the Result
  • Interpreting the Output
Inputting Your Number
Locate the input field labeled 'Number'. Enter the non-negative number for which you want to find the square root. The calculator does not support negative numbers for real square roots.
Calculating the Result
After entering the number, click the 'Calculate' button. The calculator will process the input and compute the square root.
Interpreting the Output
The result will be displayed in the 'Result' section. It will show the square root of the number you entered. For non-perfect squares, the result will be a decimal approximation.

Real-World Applications of Square Roots

  • Geometry and Engineering
  • Physics
  • Finance and Statistics
Geometry and Engineering
The Pythagorean theorem, a² + b² = c², uses square roots to find the length of a side of a right triangle. This is fundamental in architecture, construction, and engineering.
Physics
Square roots are used in many physics formulas, such as calculating the speed of an object, the period of a pendulum, or the distance between two points in a coordinate system.
Finance and Statistics
In finance, square roots are used to calculate volatility and risk. In statistics, the standard deviation, a measure of data dispersion, is calculated using a square root.

Common Misconceptions and Correct Methods

  • Square Root of a Negative Number
  • √(a + b) vs √a + √b
  • Approximation Methods
Square Root of a Negative Number
In the real number system, you cannot take the square root of a negative number. The result is an imaginary number. For example, √-1 is denoted as 'i'. This calculator only handles real numbers.
√(a + b) vs √a + √b
A common mistake is to assume that the square root of a sum is the sum of the square roots. This is incorrect. For example, √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7.
Approximation Methods
For non-perfect squares, the square root is an irrational number. Methods like the Babylonian method or Newton's method are used to find a close approximation of the square root.

Correct and Incorrect Examples

  • Correct: √(4 * 9) = √36 = 6 and √4 * √9 = 2 * 3 = 6
  • Incorrect: √(4 + 9) ≠ √4 + √9

Mathematical Derivation and Examples

  • The Babylonian Method
  • Example: Finding √10
  • Properties of Square Roots
The Babylonian Method
This is an iterative method to approximate a square root. Start with an initial guess 'g'. The next, better guess is calculated as (g + n/g) / 2, where 'n' is the number whose square root is being found. Repeat until the desired precision is achieved.
Example: Finding √10
Let n = 10. Start with a guess, say g = 3. Next guess = (3 + 10/3) / 2 = (3 + 3.33) / 2 = 3.165. Next guess = (3.165 + 10/3.165) / 2 ≈ 3.162277. The actual value is approximately 3.16227766.
Properties of Square Roots
For non-negative numbers a and b: √(ab) = √a * √b and √(a/b) = √a / √b.

Property Examples

  • √(4 * 25) = √100 = 10
  • √4 * √25 = 2 * 5 = 10