Square Root Calculator

How to Find the Square Root of Any Number

The square root calculator finds √n for any positive number. With 40,500 monthly searches, this is a fundamental math tool. The square root of n is the number that, multiplied by itself, equals n: √n = x means x² = n. √144 = 12 because 12 × 12 = 144. √2 = 1.41421356… (irrational — never terminates or repeats).

Square roots appear everywhere: the Pythagorean theorem (c = √(a² + b²)), distance formula in geometry, standard deviation (σ = √variance), and orbital mechanics (Kepler's third law). Understanding how to compute and interpret them is foundational math.

Perfect Squares Reference Chart (1–25)

• √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5
• √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10
• √121 = 11, √144 = 12, √169 = 13, √196 = 14, √225 = 15
• √256 = 16, √289 = 17, √324 = 18, √361 = 19, √400 = 20
• √441 = 21, √484 = 22, √529 = 23, √576 = 24, √625 = 25

How to Calculate Square Root by Hand

Estimation method: Find the two perfect squares your number falls between. √75 lies between √64 = 8 and √81 = 9. Since 75 is 11/17 of the way from 64 to 81: 8 + (11/17) ~ 8.65. Actual: 8.6603…

Newton's method (Babylonian): Start with a guess g, then improve: g_new = (g + n/g) / 2. For √50 starting with g = 7: iteration 1: (7 + 50/7)/2 = 7.071. Iteration 2: 7.071. Converges in just 2 iterations to 4 significant digits.

Square Root Properties and Rules

• √(a × b) = √a × √b: √72 = √(36 × 2) = 6√2
• √(a / b) = √a / √b: √(25/4) = 5/2 = 2.5
• √(a²) = |a|: Always positive, even if a is negative
• Squaring cancels square root: (√n)² = n for all n >= 0
• √(−n) is imaginary: √(−9) = 3i, where i = √(−1)
• Fractional exponent: √n = n^(1/2); cube root = n^(1/3)

Frequently Asked Questions

How do I simplify a square root?

Factor out all perfect squares. √200 = √(100 × 2) = 10√2. √48 = √(16 × 3) = 4√3. √75 = √(25 × 3) = 5√3. Find the largest perfect square factor, take its root outside the radical sign, leave the remaining factor under the radical. This "simplified radical form" is what teachers and textbooks expect as an exact answer.

What is the square root of a negative number?

Negative numbers don't have real square roots — no real number times itself gives a negative result. Instead, we use imaginary numbers: √(−n) = i√n where i = √(−1). So √(−16) = 4i. Complex numbers (a + bi) combine real and imaginary parts and are used extensively in electrical engineering, quantum mechanics, and signal processing.

Square Root Calculator: Applications in Geometry and Physics

Square roots are embedded in foundational formulas across mathematics and science. The Pythagorean theorem (c = √(a² + b²)) requires a square root to find the hypotenuse of a right triangle — fundamental in construction, navigation, and computer graphics. The distance formula between two points (x₁,y₁) and (x₂,y₂) is d = √((x₂−x₁)² + (y₂−y₁)²), extending the Pythagorean theorem to coordinate geometry.

In physics: the velocity of a wave on a string is v = √(T/μ) where T is tension and μ is linear mass density. The period of a simple pendulum is T = 2π√(L/g) where L is length and g is gravitational acceleration (9.8 m/s²). Escape velocity from Earth's surface is v_esc = √(2GM/r) ~ 11.2 km/s. Square roots appear wherever area quantities need to be converted back to linear quantities.

In statistics, standard deviation = √(variance) — the square root is essential for converting the squared unit of variance back to the original unit of measurement. Root Mean Square (RMS) values used in electrical engineering are literally the square root of the mean of squared values, giving the effective value of an alternating current or voltage. The RMS value of a sine wave is its peak value ÷ √2 ~ 0.707 × peak.

Square Root Calculator: The History of Square Roots

Square roots have been computed for at least 4,000 years. The Babylonians (circa 1800–1600 BCE) had clay tablets with remarkable approximations: they calculated √2 ~ 1.41421356, accurate to 6 decimal places. The method they used — now called the Babylonian or Newton-Raphson method — is still the most efficient iterative approach. Ancient Indian mathematicians (Sulba Sutras, circa 800 BCE) used square roots in construction geometry for making ritual altars with precise proportions.

The discovery that √2 is irrational — that it cannot be expressed as a fraction of two integers — reportedly shocked the ancient Greeks. According to legend, Hippasus of Metapontum was drowned at sea by fellow Pythagoreans for revealing this proof, which contradicted their philosophical belief that all numbers are rational. Whether or not this story is true, the proof that √2 is irrational represents a landmark in mathematical history: the recognition that numbers exist beyond the integers and fractions.

Modern computing makes square root calculation nearly instantaneous, but fast algorithms still matter for efficiency in CPU-intensive applications. The Quake fast inverse square root — a famous hack from the 1999 video game Quake III Arena — approximated 1/√x using a clever combination of integer arithmetic and Newton's method iterations, achieving a 4× speedup over the standard sqrt() function on the hardware of that era. The code contained the comment "what the f*** f*** is this?" and is now a famous piece of programming history, studied for its ingenuity.