Square Root Calculator
Find the principal and negative square roots of any real number, including perfect square detection
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Perfect Squares Reference Table
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What Is a Square Root?
A square root of a number x is a value a such that a multiplied by itself equals x (a² = x). Every positive real number has two square roots: one positive and one negative. The positive root is called the principal square root and is written as √x.
For example, the square roots of 36 are +6 and −6, because both 6² = 36 and (−6)² = 36. In most everyday contexts, when people say “the square root,” they mean the positive principal root.
Square Roots of Negative Numbers
No real number squared can produce a negative result, so the square root of a negative number is imaginary. The result uses the imaginary unit i, defined as i = √(−1). For example, √(−16) = 4i. These results are called complex or imaginary numbers and are used extensively in engineering and physics.
The Square Root Formula
The square root is equivalent to raising a number to the power of one half. This identity links square roots to exponent notation and is essential in algebra and calculus.
How to Calculate a Square Root by Hand
Calculating square roots manually is a useful skill when a calculator is not available. The simplest method for estimation uses perfect squares as reference points.
Step by Step Estimation Method
Find the two perfect squares that bracket your number. For √50, you know √49 = 7 and √64 = 8, so the answer is between 7 and 8. Since 50 is 1 unit above 49 and the gap between 49 and 64 is 15, the estimate is 7 + (1/15) ≈ 7.07. The exact value is approximately 7.0711.
For more precision, use the Babylonian method: start with an initial guess g, then repeatedly compute g = (g + x/g) / 2 until the result stabilizes. This converges very rapidly to the true square root.
Frequently Asked Questions
The square root of a number x is a value a such that a² = x. Every positive number has two square roots: a positive root and a negative root. The principal (positive) square root is written as √x. For example, √25 = 5 because 5² = 25, and the full set of square roots is ±5.
The square root of a negative number is an imaginary number. The result is expressed using the imaginary unit i, where i = √(−1). For example, √(−9) = 3i and √(−25) = 5i. These are called imaginary or complex numbers and are widely used in electrical engineering, signal processing, and quantum mechanics.
A perfect square is a positive integer whose square root is also a whole number (integer). The sequence of perfect squares starts: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. Their square roots are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 respectively.
To simplify a square root, factor the number under the radical and pull out any perfect square factors. For example, √72 = √(36 × 2) = 6√2. Find the largest perfect square that divides evenly into the number, take its square root outside the radical, and leave the remaining factor inside.
Squaring a number means multiplying it by itself (x² = x × x). Taking the square root is the inverse operation: it finds what number was squared to get x. They undo each other. For example, 7² = 49 and √49 = 7. Squaring always gives a non-negative result; square roots return the original magnitude.