Evaluate reciprocal roots using direct and iterative methods. Review steps, errors, exports, and visual trends. Perfect for classrooms, homework, coding checks, and quick verification.
The direct formula is:
Inverse Square Root = 1 / √x
For iterative refinement, the page uses Newton-Raphson on y = 1 / √x:
yn+1 = 0.5 × yn × (3 − x × yn2)
This update rapidly improves a positive starting estimate. The residual shown in the steps table checks how closely the estimate satisfies the target input.
| Input x | √x | 1 / √x |
|---|---|---|
| 0.25 | 0.5 | 2 |
| 1 | 1 | 1 |
| 2 | 1.414214 | 0.707107 |
| 4 | 2 | 0.5 |
| 9 | 3 | 0.333333 |
| 16 | 4 | 0.25 |
It is the reciprocal of a number’s square root. For any positive x, the inverse square root equals 1 divided by √x. It appears in algebra, geometry, physics, graphics, and numerical methods.
This calculator is designed for real-number results. When x is zero, division by zero occurs. When x is negative, the square root is not a real number, so the displayed formula no longer stays within real arithmetic.
The direct method computes √x first, then takes its reciprocal. It is straightforward, reliable, and useful when you want an exact numeric evaluation within ordinary floating-point precision.
The iterative method improves an initial estimate using Newton-Raphson. Each update usually sharpens the answer quickly, which makes the method helpful for learning approximation techniques and convergence behavior.
Newton-Raphson converges very fast when the starting estimate is reasonable. After only a few iterations, the estimate often matches the direct result to many decimal places, leaving a tiny absolute and relative error.
The chart plots y = 1 / √x over your selected positive range. It decreases as x grows, and the slope is steeper near small values. Submitted inputs appear as markers for quick visual comparison.
Use scientific notation when values become very small or when you want compact, consistent formatting for reports, lab work, coding checks, and exported tables.
CSV works well for spreadsheets, data cleaning, and quick sharing. PDF is better for fixed snapshots, homework submissions, project notes, and reports where layout consistency matters.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.