Square Root Calculator

The square root of a number x is the value that, when multiplied by itself, gives x back. The square root of 9 is 3, because 3 × 3 = 9. Square roots are the inverse operation of squaring, and they show up constantly in geometry (finding the side length of a square from its area), physics (calculating speed or distance from energy formulas), and statistics (standard deviation is literally defined as a square root). Every positive number has two square roots — a positive one and a negative one, since (-3) × (-3) also equals 9 — but by convention, "the square root", with the √ symbol, always refers to the non-negative one, called the principal square root. This tool always returns that principal, positive value for any non-negative input. Worked example: √64 = 8, because 8 × 8 = 64. For a non-perfect square like √20, the result is irrational and gets a long decimal expansion — approximately 4.472136 — which this tool rounds to six decimal places and trims of any trailing zeros for a clean, readable answer. Negative numbers are where square roots get genuinely more advanced. There is no real number that, multiplied by itself, produces a negative result — a negative times a negative is always positive, and a positive times a positive is also always positive. So √(-16) has no answer within the real numbers. Mathematicians solve this by defining the imaginary unit i, where i² = -1 by definition. Using that, √(-16) can be written as √16 × √(-1), or 4 × i, or "4i" for short. This tool computes exactly that: it takes the square root of the absolute value of your negative input, and appends an "i" to clearly flag that the answer is imaginary, not a plain real number. A common misconception is assuming a negative input to a square root is simply "invalid" or an error, the same way dividing by zero is. It isn't an error in the mathematical sense — it's a different, well-defined kind of number, an imaginary number, just one that doesn't behave like an ordinary decimal on a ruler or a price tag. Imaginary and complex numbers, a real part combined with an imaginary part, are essential in electrical engineering (AC circuit analysis), signal processing, and quantum mechanics, so treating √(-16) as "4i" rather than a dead end reflects how the math is actually used in practice, even if you'll rarely need it outside a classroom or specialized job. Another subtle point: √0 is exactly 0, not undefined and not imaginary — zero is its own square root, since 0 × 0 = 0, and this tool handles that boundary case the same way it handles any other non-negative number, with no special messaging needed.