Calculator Inputs
Use the expression field or separate value fields. The expression field takes priority when completed. Clear it to use separate inputs instead.
Formula Used
Negative radicands create imaginary square roots. Separate √−1 as the selected imaginary symbol. Then simplify the remaining positive fractional square root.
Perfect-square factors move outside each radical. Remaining denominator radicals are rationalised when necessary. Reduced factors create the cleanest exact final form.
How to Use
Enter one negative fraction in the expression field. Alternatively, clear it and use separate fields. Then choose output, precision, roots, and notation.
- Enter a negative numerator and nonzero denominator.
- Select principal or both square roots.
- Choose exact, decimal, or combined output.
- Select i or j notation.
- Press the calculation button.
- Copy or download the completed report.
Example Data
| Input | Principal exact result | Approximate result | Notes |
|---|---|---|---|
| √(-25/36) | 5i/6 | 0.833333i | Both terms are perfect squares. |
| √(-18/8) | 3i/2 | 1.5i | The fraction first reduces to -9/4. |
| √(-7/12) | i√21/6 | 0.763763i | The denominator radical is rationalised. |
| √(-2.25/4) | 3i/4 | 0.75i | Decimals convert into an exact fraction. |
Calculation History
History stays inside this browser. It stores recent exact and decimal results. Clearing browser data also removes saved calculations.
Frequently Asked Questions
Why does a negative fraction produce an imaginary result?
No real number squares into a negative value. Complex numbers introduce the imaginary unit i. This extension makes every negative square root meaningful.
What is the principal square root?
The principal root uses the positive imaginary direction. For √(-25/36), it equals 5i/6. Selecting both roots also displays its negative counterpart.
Can the calculator display both roots?
Yes, choose the both-roots option before calculating. The answer then receives the ± symbol. This represents positive and negative square-root solutions together.
What happens when the fraction is positive?
The calculator displays a warning about the radicand. It still computes the corresponding real square root. The imaginary component then remains exactly zero.
Can decimal values be entered?
Yes, ordinary decimal values are accepted directly. They convert into reduced exact fractions before simplification. Use nine decimal places or fewer for accuracy.
Why is the denominator rationalised?
Traditional exact notation avoids radicals within denominators. Rationalisation moves that radical into the numerator. The resulting expression is easier to compare.
What is the difference between i and j?
Both symbols represent the square root of negative one. Mathematics usually uses i for imaginary values. Engineering often uses j to avoid current confusion.
How does prime factorisation help?
Prime factors reveal every available perfect-square pair. Each pair moves outside its original radical. Unpaired factors remain inside the simplified radical expression.
Can results be downloaded?
Yes, results can download as CSV or PDF. Printing supports browser-based PDF saving as well. Optional text downloading begins after five seconds.