Complex Numbers — Operations, Polar Form & Examples

Complex numbers are numbers of the form $a+bi$, where $a$ and $b$ are real numbers and $i^2=-1$. They matter because they let you solve equations like $x^2+1=0$, and they also give a clean way to describe size and direction together.

If you only need the quick takeaway: standard form $a+bi$ is best for addition and subtraction, while polar form is often better for multiplying, dividing, and taking powers of nonzero complex numbers.

What Is A Complex Number?

In $z=a+bi$, the number $a$ is the real part and $b$ is the imaginary part. If $b=0$, then $z$ is just a real number. If $a=0$ and $b \ne 0$, the number is called purely imaginary.

It helps to picture $a+bi$ as the point $(a,b)$ in the complex plane. The real part gives the horizontal position, and the imaginary part gives the vertical position.

Some equations have no real solution but do have complex solutions. For example,

$$x^2+1=0$$

has no real solution, but in the complex numbers its solutions are $x=\pm i$.

How To Add, Subtract, Multiply, And Divide Complex Numbers

Addition and subtraction are component-wise. Combine real parts with real parts, and imaginary parts with imaginary parts:

$$(a+bi)+(c+di)=(a+c)+(b+d)i$$ $$(a+bi)-(c+di)=(a-c)+(b-d)i$$

Multiplication uses the distributive law and the fact that $i^2=-1$:

$$(a+bi)(c+di)=(ac-bd)+(ad+bc)i$$

Division is usually handled with a conjugate. The conjugate of $c+di$ is $c-di$. If $c+di \ne 0$, then

$$\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{c^2+d^2}$$

Multiplying by the conjugate removes the imaginary part from the denominator, which is why the method works.

How Polar Form Of A Complex Number Works

For a nonzero complex number $z=a+bi$, the modulus is

$$|z|=\sqrt\{a^2+b^2\}$$

The modulus is the distance from the origin to the point $(a,b)$ in the complex plane.

If $\theta$ is an angle that points to the same location as $(a,b)$, then

$$z=|z|(\cos\theta+i\sin\theta)$$

This is the polar form of $z$. The angle $\theta$ is called an argument of $z$.

The argument is not unique. If $\theta$ works, then $\theta+2\pi k$ also works for any integer $k$. Many courses choose one principal argument by convention, so check which angle range your class is using.

Polar form is useful because multiplication has a clean pattern. For nonzero complex numbers, the moduli multiply and the arguments add.

Worked Example: Multiply In Standard Form And Polar Form

Take

$$z=1+\sqrt{3}i$$

and

$$w=\sqrt{3}+i$$

First multiply in standard form:

$$zw=(1+\sqrt{3}i)(\sqrt{3}+i)$$ $$=\sqrt{3}+i+3i+\sqrt{3}i^2$$ $$=\sqrt{3}+4i-\sqrt{3}=4i$$

Now switch to polar form.

For $z=1+\sqrt{3}i$, the modulus is

$$|z|=\sqrt\{1^2+(\sqrt\{3\})^2\}=2$$

and the point $(1,\sqrt{3})$ has argument $\theta=\pi/3$. So

$$z=2\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$$

For $w=\sqrt{3}+i$, the modulus is also $2$, and the point $(\sqrt{3},1)$ has argument $\pi/6$. So

$$w=2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$$

Multiply the polar forms:

$$zw=4\left(\cos\left(\frac{\pi}{3}+\frac{\pi}{6}\right)+i\sin\left(\frac{\pi}{3}+\frac{\pi}{6}\right)\right)$$ $$=4\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)=4i$$

Both methods give the same answer. The point of the polar method is not that it is always shorter for small numbers, but that it makes the multiplication rule easy to see: lengths multiply, angles add.

Common Mistakes With Complex Numbers

The most common mistake is forgetting that $i^2=-1$. That sign change is what turns a product into the correct real part and imaginary part.

Another common mistake is choosing the wrong argument in polar form. A reference angle by itself is not enough; you also need the correct quadrant.

Students also sometimes try to add complex numbers in polar form by adding moduli and arguments. That does not work. Polar form mainly simplifies multiplication, division, and powers.

One more edge case matters: the zero complex number has modulus $0$, but its argument is not defined in the usual way. So polar form is mainly used for nonzero complex numbers.

When Complex Numbers Are Used

Complex numbers are used to solve polynomial equations, describe rotations and oscillations, and model systems in engineering and physics. They appear in AC circuits, signal processing, control theory, and quantum mechanics.

Even if you first meet them in algebra, they are not just a formal trick. They give a compact way to describe patterns that involve both magnitude and angle.

Try Solving A Similar Problem

Try writing $-1+i$ in polar form. Find its modulus, choose the argument from the correct quadrant, then square the result and convert back to standard form to check your work.