Complex Numbers for Beginners: i, Arithmetic and the Argand Diagram

A complex number has the form $a + bi$, where $i$ is defined by $i^2 = -1$. You add and multiply them like algebra, replacing $i^2$ with $-1$, and divide by multiplying top and bottom by the conjugate. On an Argand diagram, $a + bi$ is the point $(a, b)$.

Why we need $i$

$x^2 = -1$ has no real solution, since squares of real numbers are never negative. Defining $i = \sqrt{-1}$ gives every quadratic equation two solutions, such as $x^2 + 2x + 5 = 0 \Rightarrow x = -1 \pm 2i$.

Adding and subtracting

Combine real parts and imaginary parts separately:

Multiplying

Expand and use $i^2 = -1$:

Dividing

Multiply top and bottom by the conjugate of the bottom (change the sign of its imaginary part):

Modulus and argument

• Modulus $|z| = \sqrt{a^2 + b^2}$: the distance from 0. For $3 + 4i$ it is 5.
• Argument $\arg z$: the angle from the positive real axis, about 0.927 rad (53.13°) for $3 + 4i$.

Together they give the polar form $z = r(\cos\theta + i\sin\theta)$.

Common mistakes

• Treating $i^2$ as $+1$.
• Dividing real and imaginary parts separately without using the conjugate.
• Getting the argument in the wrong quadrant.

Practice questions

1. $(2 + i) + (3 - 4i)$
2. $(1 + i)^2$
3. $|5 - 12i|$

Answers: 1) $5 - 3i$ 2) $2i$ 3) 13

The complex number calculator does all four operations exactly and plots the numbers on an Argand diagram. Complex roots appear in the quadratic equation solver; see what the discriminant tells you.

Frequently asked questions

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