Complex Root Calculator
Calculate the square root of any complex number instantly. Enter the real and imaginary parts to get the exact real and imaginary components of the result, free online.
How to Use the Complex Square Root Calculator
- Enter the real part (a) of your complex number.
- Enter the imaginary part (b) of your complex number.
- Click the “Calculate” button.
- View the real part, imaginary part, and full result of the square root instantly.
Features of the Complex Square Root Calculator
- Instantly calculates the principal square root of any complex number.
- Displays the real part, imaginary part, and combined result separately.
- Works with positive, negative, and zero values for both parts.
- Correctly handles pure real and pure imaginary numbers.
- Free to use with no sign-up required.
Complex Square Root Formula
For a complex number z = a + bi, the principal square root is calculated using the modulus r = √(a² + b²). The real and imaginary parts of the root are then found without needing trigonometry:
Real part of root = √((r + a) / 2)
Imaginary part of root = sign(b) × √((r − a) / 2)
Where sign(b) is +1 if b is positive or zero, and −1 if b is negative. Squaring the result always returns the original complex number a + bi.
Common Complex Number Square Roots (Reference Table)
| Complex Number (a + bi) | Square Root |
|---|---|
| 3 + 4i | 2 + 1i |
| 5 + 12i | 3 + 2i |
| −3 + 4i | 1 + 2i |
| 0 + 8i | 2 + 2i |
| 8 + 6i | 3 + 1i |
| −5 + 12i | 1.4142 + 4.2426i |
| 4 + 0i | 2 + 0i |
| −9 + 0i | 0 + 3i |
How to Calculate the Square Root of a Complex Number
To find the square root of a complex number a + bi by hand, first calculate the modulus r = √(a² + b²). Then use r together with a to solve for the real and imaginary parts of the root, checking the sign of b to determine whether the imaginary part of the root is positive or negative.
Suppose you want the square root of 3 + 4i. First, r = √(3² + 4²) = √25 = 5. Then the real part = √((5 + 3) / 2) = √4 = 2, and the imaginary part = √((5 − 3) / 2) = √1 = 1.
Thus, the square root of 3 + 4i is 2 + 1i.
Solved Examples on Complex Square Roots
1. Square Root of a Positive Real and Imaginary Part
Example: Find the square root of 5 + 12i.
r = √(5² + 12²) = √169 = 13. Real part = √((13 + 5) / 2) = √9 = 3. Imaginary part = √((13 − 5) / 2) = √4 = 2. Result: 3 + 2i.
2. Square Root of a Negative Real Part
Example: Find the square root of −3 + 4i.
r = √((−3)² + 4²) = √25 = 5. Real part = √((5 + (−3)) / 2) = √1 = 1. Imaginary part = √((5 − (−3)) / 2) = √4 = 2. Result: 1 + 2i.
3. Square Root of a Pure Imaginary Number
Example: Find the square root of 0 + 8i.
r = √(0² + 8²) = √64 = 8. Real part = √((8 + 0) / 2) = √4 = 2. Imaginary part = √((8 − 0) / 2) = √4 = 2. Result: 2 + 2i.
4. Square Root of a Negative Imaginary Part
Example: Find the square root of 8 − 6i.
r = √(8² + (−6)²) = √100 = 10. Real part = √((10 + 8) / 2) = √9 = 3. Imaginary part = −√((10 − 8) / 2) = −1. Result: 3 − 1i.
1. What is the square root of a complex number?
The square root of a complex number a + bi is another complex number that, when multiplied by itself, gives back a + bi. Every non-zero complex number has two square roots, but this calculator returns the principal (positive-oriented) root.
2. Why does a complex number have two square roots?
Just like real numbers, complex numbers have two square roots that are negatives of each other. If x is a square root of a + bi, then −x is also a square root, since (−x)² equals x².
3. Can the square root of a negative real number be found this way?
Yes. Setting the imaginary part to 0 lets you find the square root of any negative real number, such as −9, which returns 0 + 3i, since 3i squared equals −9.
4. How do you verify a complex square root is correct?
Square the result and confirm it matches the original number. For example, squaring 2 + 1i gives 4 + 4i − 1 = 3 + 4i, which matches the original input.
5. Does the calculator work with decimal values?
Yes, the calculator accepts decimal and negative values for both the real and imaginary parts, and returns results rounded to four decimal places.
6. Where are complex square roots used in practice?
Complex square roots are used in electrical engineering for impedance calculations, in signal processing, and in solving polynomial equations that have no real solutions.