Complex Numbers — Operations, Polar Form & Examples

Complex numbers are numbers of the form $a+bi$, where $a$ and $b$ are real numbers and $i^2=-1$. The defining symbol is $i$, the imaginary unit, fixed by the single rule

That one rule is what lets you solve equations like $x^2+1=0$, and it gives a clean way to describe size and direction together. In $z=a+bi$, the number $a$ is the real part and $b$ is the imaginary part.

Why The Rule $i^2=-1$ Makes The System Work

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

has no real solution, but in the complex numbers its solutions are $x=\pm i$, precisely because squaring $i$ produces $-1$. Geometrically, 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. If $b=0$, then $z$ is just a real number; if $a=0$ and $b \ne 0$, the number is purely imaginary.

Arithmetic In Standard Form

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

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

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

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

Polar Form, And Why It Helps

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

the distance from the origin to the point $(a,b)$. If $\theta$ is an angle pointing to the same location as $(a,b)$, then

This is the polar form, and $\theta$ is called an argument of $z$. The argument is not unique: if $\theta$ works, then $\theta+2\pi k$ works for any integer $k$, so check which principal range your class uses. Polar form is useful because, for nonzero numbers, multiplication has a clean pattern: the moduli multiply and the arguments add.

Worked Example: Multiply In Both Forms

Take

First multiply in standard form:

Now switch to polar form. For $z=1+\sqrt{3}i$, the modulus is

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

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

Multiply the polar forms:

Both methods give $4i$. The point of the polar method is not that it is shorter for small numbers, but that it makes the rule visible: lengths multiply, angles add.

Practice And Sanity Checks

Write $-1+i$ in polar form: find its modulus, choose the argument from the correct quadrant, then square the result and convert back to standard form to verify. While you work, keep these traps in mind:

• The most common error is forgetting that $i^2=-1$. That sign change is what produces the correct real and imaginary parts.
• A reference angle alone is not enough in polar form; you also need the correct quadrant.
• You cannot add complex numbers in polar form by adding moduli and arguments. Polar form mainly simplifies multiplication, division, and powers.
• The zero complex number has modulus $0$, but its argument is undefined in the usual way, so polar form is used for nonzero numbers.

Complex numbers are not just a formal trick. They solve polynomial equations and describe rotations and oscillations, appearing in AC circuits, signal processing, control theory, and quantum mechanics whenever both magnitude and angle matter.