Mathematics · Calculus
Newton Square-Root Iteration Calculator
Perform one Newton iteration toward the square root of a positive number.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- x_next=(3+10÷3)÷2=3.166666666666667.
- Squared error changes from -1 to 0.027777777777780344.
Problem → model → reason → result
What problem does this model solve?
Perform one Newton iteration toward the square root of a positive number.
Why does the model apply?
Newton's method replaces the current guess by the tangent line's root.
What assumptions does it make?
The function follows the displayed form and is differentiable at the evaluation point. The local rate produced by a derivative is not automatically an average rate over an interval.
Formula
x_next=(x+S/x)/2
Calculation and working
The calculator above substitutes your inputs into the model and leaves the calculation steps visible so the answer can be checked rather than merely accepted.
What does the result mean?
The result answers the stated problem in the units implied by your inputs. Read its sign, size and units together, then compare it with the original values before drawing a conclusion.
Worked example
For S=10 and guess 3, the next guess is 3.1667.
Common mistake
A zero starting guess causes division by zero; multiple steps may be needed.
When does this model not apply?
This compact symbolic model covers the displayed function family, not every function, discontinuity or domain restriction.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Newton square-root step do?
Perform one Newton iteration toward the square root of a positive number.
How does the Newton square-root step work?
The calculator applies x_next=(x+S/x)/2. Newton's method replaces the current guess by the tangent line's root.
What can I learn from the Newton square-root step?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.