Calculator Form
Formula Used
For negative values: √(-a) = i√a, where a > 0
This means a negative number does not have a real square root. Its square root is imaginary and uses the symbol i, where i² = -1.
For example: √(-49) = i√49 = 7i
If the inside value contains a square factor, the simplified form can be reduced.
Example simplification: √(-18) = i√18 = i√(9×2) = 3√2i
How to Use This Calculator
- Enter the radicand, usually a negative number such as -25.
- Choose the decimal precision you want for approximations.
- Select whether you want decimal form, simplified form, or both.
- Choose principal root only or both roots together.
- Pick standard notation, scientific notation, or both styles.
- Click the calculate button to see the result, graph, and export options above the form.
Example Data Table
| Radicand | |Radicand| | Principal Root | Both Roots | Simplified Form |
|---|---|---|---|---|
| -1 | 1 | i | ±i | i |
| -4 | 4 | 2i | ±2i | 2i |
| -9 | 9 | 3i | ±3i | 3i |
| -18 | 18 | 4.242641i | ±4.242641i | 3√2i |
| -50 | 50 | 7.071068i | ±7.071068i | 5√2i |
| -0.25 | 0.25 | 0.5i | ±0.5i | Use decimal form |
FAQs
1. What is the square root of a negative number?
A negative number has no real square root. Its square root is imaginary and uses i, where i² equals -1. For example, √(-9) becomes 3i.
2. Why does the calculator return an imaginary answer?
It returns an imaginary answer because squaring any real number cannot produce a negative result. Imaginary numbers extend the number system so roots of negative values can be expressed correctly.
3. What does the principal root mean?
The principal root is the main square root value chosen by convention. For a negative radicand, the principal square root lies on the positive imaginary axis, such as √(-16) = 4i.
4. Why are there two roots for the same radicand?
Squaring both a positive value and its negative gives the same radicand magnitude. So the roots occur as opposites, such as ±4i for -16, even though the principal root is usually shown first.
5. Can this calculator simplify radicals like √(-18)?
Yes. When the absolute value contains square factors, the calculator simplifies them. For instance, √(-18) becomes 3√2i because 18 = 9×2 and √9 can be moved outside the radical.
6. Does the tool work for decimals?
Yes. Decimal radicands are supported. The calculator provides decimal approximations for values like -0.25, giving 0.5i. Exact simplified radicals are mainly useful when the absolute value is an integer.
7. What does the verification line show?
The verification line squares the displayed principal root to confirm the original radicand. It helps you check that the computed imaginary or real value is mathematically consistent.
8. When should I use scientific notation here?
Scientific notation is helpful for very large or very small values. It keeps the result compact, readable, and easier to compare when standard decimal formatting becomes too long.