Square Root
Calculate the square root of any non-negative number instantly with this free online square root calculator. Get precise sqrtx results with up to 6 decimal places for students, teachers, engineers, and professionals.
About This Calculator
What is a Square Root?
A square root of a number x is a number y such that y x y = x, written as sqrtx = y. For example, the square root of 16 is 4 because 4 x 4 = 16. The square root is the inverse operation of squaring a number. Every positive real number has two square roots: a positive root (denoted sqrtx) and a negative root (denoted -sqrtx). The principal square root is the non-negative root.
This free online square root calculator provides instant, accurate results for any non-negative number you enter. Simply type a number into the input field and click Calculate to get the precise square root value with up to 6 decimal places. The calculator also supports decimal numbers and zero.
How the Square Root Formula Works
Mathematically, the square root is expressed using the radical symbol sqrt or as a fractional exponent: sqrtx = x1/2 = x0.5. This notation works because (x0.5)2 = x0.5 x 2 = x1 = x. The square root function f(x) = sqrtx is defined for all non-negative real numbers and returns the principal (non-negative) square root.
Common Square Roots to Remember
- sqrt1 = 1
- sqrt4 = 2
- sqrt9 = 3
- sqrt16 = 4
- sqrt25 = 5
- sqrt36 = 6
- sqrt49 = 7
- sqrt64 = 8
- sqrt81 = 9
- sqrt100 = 10
- sqrt121 = 11
- sqrt144 = 12
Real-World Applications
Square roots appear in many fields. In geometry, the square root gives the side length of a square from its area and appears in the Pythagorean theorem (c = sqrt(a^2 + b^2)). In physics, square roots are used in formulas for kinetic energy, pendulum period, and standard deviation. In finance, they are essential for calculating volatility, risk metrics, and compound growth rates. Engineers use square roots in signal processing, structural analysis, and electrical circuit design.
Frequently Asked Questions
What is a square root?
A square root of a number x is a number y such that y multiplied by itself (y x y) equals x. It is written as sqrtx = y. For example, the square root of 16 is 4 because 4 x 4 = 16. Every positive real number has two square roots: a positive and a negative root.
Can you find the square root of a negative number?
The square root of a negative number is not a real number but an imaginary or complex number. For example, sqrt(-9) = 3i, where i = sqrt(-1) is the imaginary unit. This calculator returns results only for non-negative real numbers. Complex square roots involve imaginary numbers and are used in advanced mathematics, physics, and engineering.
What is the difference between square root and cube root?
A square root finds a number that when multiplied by itself gives the original value (y^2 = x). A cube root finds a number that when multiplied by itself three times gives the original value (y^3 = x). Unlike square roots, cube roots exist for all real numbers including negatives. Zero's square root and cube root are both zero.
How do you calculate square roots without a calculator?
You can estimate square roots by finding the nearest perfect squares and interpolating. For example, sqrt52 is between sqrt49 = 7 and sqrt64 = 8, and since 52 is closer to 49, sqrt52 ≈ 7.2. For more precision, use the Babylonian method: start with an estimate, divide your number by it, average the result with the estimate, and repeat.
What are perfect squares?
A perfect square is a number that is the square of an integer. For example, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 are perfect squares because they equal 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2, 11^2, and 12^2 respectively. Numbers that are not perfect squares have irrational square roots.
Is the square root of 2 a rational number?
No, the square root of 2 (sqrt2 ≈ 1.41421) is an irrational number. It cannot be expressed as a fraction of two integers. This was proven by ancient Greek mathematicians and is one of the earliest known proofs of irrationality. Most square roots of non-perfect squares are irrational numbers.
What are real-world applications of square roots?
Square roots are used in geometry to find side lengths of squares from area, in physics for calculating velocity from kinetic energy and in the Pythagorean theorem, in engineering for stress analysis and signal processing, in finance for standard deviation and volatility calculations, and in statistics for root mean square error and confidence intervals.