Enter any number into the Number (x) field and the Square Root Calculator instantly computes √x — the principal square root — along with the negative square root, the squared value (x²), and whether your number is a perfect square. Works with decimals, large numbers, and negative inputs (returning imaginary results).
Square Root (√x)
--
Negative Square Root (−√x)
--
Squared Value (x²)
--
Perfect Square?
--
Yes. Every positive real number has two square roots: a positive (principal) root and a negative root. For example, √9 = 3 and −√9 = −3, because both 3² and (−3)² equal 9. By convention, the square root symbol (√) refers to the principal (positive) root.
A perfect square is an integer whose square root is also a whole integer. Examples include 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. If √x results in an exact whole number, x is a perfect square.
One common manual method is the estimate-and-average approach: guess a number b, divide x by b to get c, then average b and c as your new guess. Repeat until the desired precision is reached. For example, to find √27, start with a guess of 5, then refine: 27÷5=5.4, average gives 5.2, and so on.
Find the two perfect squares your number falls between. For example, √50 lies between √49=7 and √64=8, so the answer is between 7 and 8. Since 50 is close to 49, you can estimate √50 ≈ 7.07. This method gives a quick rough answer without a calculator.
No. √2 is an irrational number, meaning it cannot be expressed as a simple fraction. Its decimal expansion (≈ 1.41421356…) goes on forever without repeating. In fact, the square root of any non-perfect-square integer is irrational.
Negative numbers do not have real square roots because no real number squared gives a negative result. Instead, the square root of a negative number is expressed as an imaginary number using the unit i, where i = √(−1). For example, √(−9) = 3i.
To eliminate a square root, square both sides of the equation. For example, if √x = 5, squaring both sides gives x = 25. Always check your solution afterwards, as squaring can introduce extraneous (false) solutions.
The word 'radical' comes from the Latin word 'radix,' meaning root. The symbol √ evolved from a stylized lowercase 'r' used by medieval mathematicians. The horizontal bar extending over the number (the vinculum) was added later to group the expression under the root.