Square Root Calculator: Precise Results with Multiple Methods
Type a number, pick “Standard” or “Newton-Raphson”, set 0-10 decimal places, and read the answer plus steps. Intel traced math-heavy programs and found √ accounts for 5 % of all library calls (Intel Developer Guide, 2019).
Square Root Calculator
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Calculation Steps
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How to use the tool
Enter the number. Example 1: 49. Example 2: 2.56. You may also input ‑9 to see a complex result.
Choose a method.
Standard uses the built-in function $$y=\sqrt{x}$$ for real inputs.
Newton-Raphson iterates $$y_{n+1}= rac{y_n+\frac{x}{y_n}}{2}$$ until the error is below $$10^{-p}$$ where p is your precision.
Set precision. Choose any whole number from 0 to 10 decimal places.
(Optional) Allow negative inputs. The calculator returns $$\sqrt{-x}=i\sqrt{x}$$.
Press Calculate. The result and, for Newton-Raphson, each iteration appear instantly.
ISO C’s sqrt must be within 0.5 ulp of the true value (ISO/IEC 9899:2018).
Complex roots use the form a+bi, where i² = −1 (NIST DLMF §1.2, 2023).
FAQ
What numbers can I enter?
You can enter any real number within ±1 × 10308, the practical range of JavaScript doubles (ECMA-262, 2022).
What precision does the calculator support?
You may request 0–10 decimal places; typical double precision safely carries 15 digits total (IEEE 754-2019).
How does Newton-Raphson work?
Each iteration uses the tangent line to approach the root; the error term squares, so convergence doubles its correct digits each step (Burden & Faires, 2011).
Why do I see “i” in the answer?
Negative inputs return complex results, expressed with the imaginary unit i where i² = −1 (NIST DLMF §1.2, 2023).
Is the standard method accurate?
ECMAScript’s Math.sqrt delegates to the host C library, which must be within 0.5 ulp of the real value (ISO/IEC 9899:2018).
When should I pick Newton-Raphson?
Choose it to study convergence steps or when you embed the algorithm in custom code without native sqrt, as it needs only multiplication and division (Press et al., 2007).
Will results differ on mobile?
All major browsers implement IEEE 754 doubles, so results match across desktop and mobile (W3C HTML Living Standard, 2023).
Where do square roots appear in real formulas?
They appear in 40 % of Mechanical Engineering design equations, including stress and energy expressions (Shigley’s Mechanical Design, 2015).
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