Square Root Calculator (√n)

√n with integer-perfect-square check, simplified surd form, and decimal expansion.

Inputs

Result

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General calculation reads

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How Not to Be Wrong
Jordan Ellenberg - math thinking
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Steven Strogatz - accessible math
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Scientific calculator (Casio fx-991EX)
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How to use this calculator

  • Enter non-negative number.
  • Read decimal + simplified surd form.

About this calculator

The square root of n is the value x where x² = n. Perfect squares (1, 4, 9, 16, 25, …) have integer roots; everything else has irrational roots. Simplified surd form pulls out the largest perfect-square factor: √50 = √(25 × 2) = 5√2. JavaScript Math.sqrt is implementation-defined but typically uses hardware FPU instruction (CVTSS2SS, FSQRT) — accurate to last bit. For huge integers, Newton's method or arbitrary-precision libraries.

Frequently asked

Why simplified surds?+
Cleaner exact representation. √50 = 5√2 — easier to reason about than 7.0710678…
Negative numbers?+
Real sqrt undefined; complex sqrt is i × √|n| (imaginary). This calc returns "imaginary" notice.
How is sqrt computed?+
Hardware FSQRT (Intel/ARM): typically Newton-Raphson. JavaScript exposes Math.sqrt; returns IEEE 754 nearest-rounded.
Big integers?+
JS Number gives 15 digits. For larger, use BigInt + Newton iteration. This calc covers most needs to ~10^15.
Alternative: nth-root?+
Generalized: x^(1/n). Square root = x^(1/2). Cube root = x^(1/3). See cube-root for ∛.

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