Calculator Inputs
The default number is 123. You may test another non-negative value.
Formula Used
How to Use
Enter 123 or another non-negative number. Select the required square-root sign. Choose your preferred approximation method.
Set numerator and denominator limits. Choose precision and rounding controls. Add an acceptable error when needed.
Press calculate to view results. Compare convergents and denominator limits. Copy, print, or download your calculation.
Example Data
| Maximum denominator | Approximate fraction | Approximate decimal |
|---|---|---|
| 10 | 111/10 | 11.1 |
| 100 | 122/11 | 11.09090909 |
| 1,000 | 2695/243 | 11.09053498 |
| 10,000 | 29767/2684 | 11.09053651 |
Calculation History
Frequently Asked Questions
Can square root of 123 be an exact fraction?
No. Square root of 123 is irrational. Any ordinary fraction is only an approximation.
Why does the radical remain unchanged?
The number 123 has no square factor. Its prime factors are three and forty-one.
What does maximum denominator mean?
It limits the denominator used during searching. Larger limits usually create more accurate fractions.
What is a continued fraction?
It expresses a number through nested reciprocals. Its convergents often provide excellent rational approximations.
What is absolute error?
Absolute error measures the direct numerical difference. Smaller values indicate a closer approximation.
What is relative error?
Relative error compares absolute error with the true value. It helps compare different calculations fairly.
Can I request a fraction below the root?
Yes. Select the below option. The result will not exceed the square root.
Can this calculator process other numbers?
Yes. Replace 123 with another non-negative value. Whole numbers receive extra radical analysis.
Why do numerator limits matter?
They prevent excessively large fractions. This improves performance and keeps displayed answers practical.