Square Root by Hand Calculator
This square root by hand calculator finds the square root of a positive number using Newton's method, the classic iterative technique you can carry out with nothing but multiplication and division. The idea is simple and powerful: start with any rough guess for the root, then repeatedly replace the guess with the average of the guess and the number divided by the guess. Each pass roughly doubles the number of correct digits, so a handful of steps reaches full precision. This method, also called the Babylonian method, is the standard root-finding procedure described in numerical references such as those from the US National Institute of Standards and Technology, and it is how many calculators compute roots internally. Enter the number you want the root of and the calculator shows the converged value to six decimal places, the square you can check it against, and the number of iterations used. Use it to understand how square roots are computed, to check a result by hand, or to teach iterative methods. Every figure is computed deterministically from the Newton iteration shown in full below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
Newton's method averages a guess with the number divided by the guess: starting from 1, the square root of 2 converges to 1.414214 in a few steps, since 1.414214 squared is 2.000000. Each iteration roughly doubles the correct digits.
Newton's square root iteration
x_(k+1) = ( x_k + n / x_k ) / 2
x_0 = initial guess (here 1)
n = number whose root is sought
repeat until x stops changing to the wanted precision
Each step takes the current estimate, divides the target by it, and averages the two. If the estimate is too small, n divided by it is too large, so their average lands closer to the true root. The error squares at every step, which is why convergence is so fast.
Worked example
Find the square root of 2 starting from a guess of 1.
- x1 = (1 + 2 / 1) / 2 = 1.500000
- x2 = (1.5 + 2 / 1.5) / 2 = 1.416667
- x3 = (1.416667 + 2 / 1.416667) / 2 = 1.414216
- x4 = (1.414216 + 2 / 1.414216) / 2 = 1.414214
- 1.414214 squared = 2.000000, so the root is 1.414214
The square root of 2 converges to 1.414214, and squaring it returns 2.000000. These are the calculator's default inputs, so the result above matches the widget exactly.
Convergence of Newton's method for the square root of 2
Starting from 1, each iteration roughly doubles the number of correct decimal digits.
| Iteration | Estimate |
|---|---|
| 0 | 1.000000 |
| 1 | 1.500000 |
| 2 | 1.416667 |
| 3 | 1.414216 |
| 4 | 1.414214 |
Numerical methods reference: US National Institute of Standards and Technology (NIST).
Square Root by Hand Calculator: frequently asked questions
What is Newton's method for square roots?
It is an iterative formula that refines a guess for the square root of n by replacing it with the average of the guess and n divided by the guess. Written as x next equals (x + n/x)/2, it converges quadratically, meaning the number of correct digits roughly doubles each step. It is also known as the Babylonian method because it was used in antiquity.
Why start the guess at 1?
Any positive starting value works because the method converges from almost any guess. Starting at 1 keeps the worked example simple and shows the convergence clearly. A guess closer to the true root reaches full precision in fewer steps, which is why production code often seeds the guess using the exponent of the number.
How many iterations are needed?
Because the error squares each step, six to eight iterations are usually enough to reach the limit of double-precision arithmetic for any ordinary number. This calculator iterates until the estimate stops changing to its working precision, then reports how many steps it took, typically a single-digit count.
Can it handle non-perfect squares?
Yes. The method works for any non-negative real number, not just perfect squares. The square root of 2 is irrational and never terminates, yet Newton's method still converges to it quickly; the calculator simply shows the value rounded to six decimal places.
What happens with zero or a negative number?
The square root of zero is zero, which the calculator returns directly. Negative numbers have no real square root, so the calculator does not produce a real result for them; complex roots are outside the scope of this real-valued tool.
Official sources
- Numerical methods and mathematical functions reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.