Number Systems → Complex Numbers — Learn by Doing

Number Systems — Definitions & Examples

The sets nest like this: \(\mathbb N\subseteq\mathbb W\subseteq\mathbb Z\subseteq\mathbb Q\subseteq\mathbb R\subseteq\mathbb C\).

Natural Numbers (\(\mathbb N\))

Counting numbers. Many syllabi use \(\{1,2,3,\dots\}\); some include 0. We’ll use convention \(\mathbb N=\{1,2,3,\dots\}\).

Whole Numbers (\(\mathbb W\))

All natural numbers and 0.

Integers (\(\mathbb Z\))

Whole numbers and their negatives.

Rational Numbers (\(\mathbb Q\))

Fractions of integers (decimals terminate or repeat).

Irrational Numbers (\(\mathbb R\setminus\mathbb Q\))

Decimals neither terminate nor repeat (e.g. \(\pi,e,\sqrt{2}\)).

Real & Complex

\(\mathbb R\) are all points on the number line; \(\mathbb C\) are \(a+bi\) with \(i^2=-1\).

Complex Numbers — Core Concepts

1) Basics & Algebra with \(a+bi\)

A complex number is \(z=a+bi\) where \(i^2=-1\). Real part \(\Re(z)=a\), imaginary part \(\Im(z)=b\), conjugate \(\overline z=a-bi\).

2) Modulus and Argument

\(|z|=\sqrt{a^2+b^2}\); \(\arg(z)=\theta\) via atan2. Also \(z\overline z=|z|^2\).

3) Polar & Euler Form

\(z=r(\cos\theta+i\sin\theta)=re^{i\theta}\)

4) De Moivre & Roots

\(z_k = r^{1/n} e^{i(\theta+2k\pi)/n}\) for \(k=0,\dots,n-1\).

Complex Numbers Calculator

Argand Diagram — Plot & Transform

Powers & n-th Roots Visualizer

Enter \(z=r e^{i\theta}\) then view \(z^n\) and the \(n\) roots.

Common Loci on the Argand Plane

• Circle: \(|z-a|=R\)
• Perpendicular Bisector: \(|z-z_1|=|z-z_2|\)
• Half-plane: \(\Re(\alpha z)\ge c\)

Quizzes

Number Systems Quiz

Complex Numbers Quiz