Tip: For negative numbers, this tool also shows the complex-number form.
What this square root calculator does
This calculator finds the square root of any number you enter. For positive numbers, it returns the principal square root, which is the non-negative result of √x. It also shows the second real solution (negative value) when relevant, because if x is positive, both +r and -r satisfy x = r².
For negative inputs, there is no real square root, so the calculator displays the equivalent complex number form using i.
How to use the calculator
- Type a number in the input field (integer or decimal).
- Choose how many decimal places you want (0 to 15).
- Click Calculate √x (or press Enter).
- Read the principal root and supporting explanation in the result box.
Square roots explained simply
Definition
The square root of a number n is a value r such that r × r = n. In notation, r = √n. For example, √49 = 7 because 7 × 7 = 49.
Principal root vs. equation solutions
The symbol √n usually means the principal (non-negative) root. But the equation x² = n has two real solutions when n > 0: x = +√n and x = -√n.
Common perfect squares
- √0 = 0
- √1 = 1
- √4 = 2
- √9 = 3
- √16 = 4
- √25 = 5
- √36 = 6
- √49 = 7
- √64 = 8
- √81 = 9
- √100 = 10
Examples
Example 1: Perfect square
Input: 144. Output: 12 exactly. Since 12² = 144, the principal square root is 12.
Example 2: Decimal input
Input: 2.25. Output: 1.5 exactly. This is useful for scaling, geometry, and data normalization.
Example 3: Non-perfect square
Input: 2. Output: approximately 1.414214 (at 6 decimal places). The decimal continues infinitely.
Where square roots are used in real life
- Geometry: finding side lengths and diagonal distances.
- Statistics: standard deviation calculations.
- Physics: formulas involving area, energy, and wave behavior.
- Finance: volatility and risk measures in quantitative models.
- Engineering: signal processing and optimization problems.
FAQ
Can I take the square root of a negative number?
Not in the real number system. In complex numbers, √(-a) = i√a, where i² = -1.
How precise is this calculator?
It uses JavaScript's built-in floating-point math. For everyday use, it is very accurate. You can set your preferred decimal precision from 0 to 15 places.
Why does 0 only have one square root?
Because +0 and -0 are the same numeric value. So x² = 0 has one unique solution: x = 0.