Understanding square roots of a complex number

Expert intro: This definition-based topic is important because complex numbers behave differently from real numbers, yet the square-root idea still follows a precise algebraic rule.

Detailed walkthrough

Definition of a square root in the complex plane

A square root of a complex number $z$ is any complex number $w$ such that

$w^2=z.$

That is the entire definition. In the complex numbers, we are not limited to real values, so a number can have square roots even when it is negative or non-real.

Why a nonzero complex number has two square roots

If $z\neq 0$, then it has exactly two square roots. This happens because the equation

$w^2=z$

is quadratic in $w$. A quadratic equation over the complex numbers has two solutions, counted with multiplicity. Geometrically, if one root is $w$, then the other is $-w$, and both square to the same number because

$(-w)^2=w^2=z.$

So the two square roots are negatives of each other.

Example with a simple complex number

Take $z=-1$. The square roots are $i$ and $-i$, because

$i^2=-1 \quad\text{and}\quad (-i)^2=-1.$

This illustrates the main idea: complex square roots often come in pairs, and they are not restricted to real numbers.

Important properties to remember

For $z\neq 0$, the two square roots are always opposite numbers. If $z=0$, then the only square root is $0$ itself, because $w^2=0$ implies $w=0$.

When solving more advanced problems, it is often useful to write a complex number in polar form $z=re^{i\theta}$. Then its square roots are

$\sqrt r\,e^{i\theta/2} \quad\text{and}\quad -\sqrt r\,e^{i\theta/2},$

with the angle adjusted appropriately. But for the basic definition, all you need is the equation $w^2=z$.

Summary of the concept

The key takeaway is that a square root of a complex number is any complex number whose square equals the original number. Nonzero complex numbers always have two such roots, and they are negatives of each other.

💡 Pitfall guide

A frequent mistake is assuming that square roots must be real. That is true in some real-number settings, but not in the complex plane. Another error is thinking that a nonzero complex number has only one square root because a calculator may display one principal value. The principal square root is a convention, not the full set of roots. Also, when using polar form, students sometimes forget that halving the angle still leaves a second root obtained by multiplying by $-1$; both must be included when the problem asks for all square roots.

🔄 Real-world variant

If the question were changed to "Find the square roots of $z=3-4i$," then the definition would still be $w^2=z$, but you would need to solve for $w=a+bi$ and match real and imaginary parts. That leads to a system of equations, often with two solutions that are negatives of each other. If the number were written in polar form instead, you could use the magnitude and angle to obtain the roots more directly. The underlying concept is the same, but the solving method changes.

🔍 Related terms

principal square root, polar form, complex conjugates