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Mathematics › 🌌 Grade 11 › The Complex Plane: Argand Diagrams and Operations
🌌 Grade 11 · Lesson 11 of 12

The Complex Plane: Argand Diagrams and Operations

Give the number line a second dimension and √−1 finds a home — every complex number becomes a point, addition becomes a slide, and multiplication becomes a rotate-and-stretch.

Grade 11Algebra 2 / Pre-Calculus
The Complex Plane: Argand Diagrams and Operations — illustration
💡
The big idea: The equation x² = −1 has no real solution, so mathematicians defined a new number i with i² = −1. Combining it with real numbers gives the complex numbers a + bi, which live not on a line but on a plane: the real part along one axis, the imaginary part along the other. On this Argand diagram, adding complex numbers is vector addition, and multiplying them rotates and scales — turning algebra into geometry.
🎯 By the end, you'll be able to
  • Define the imaginary unit i and simplify powers of i
  • Plot a complex number a + bi as a point on the Argand diagram
  • Add, subtract, and multiply complex numbers in a + bi form
  • Interpret the modulus and argument, and multiplication as rotation and scaling
📎 You should already know
  • The coordinate plane
  • Square roots and the quadratic formula

A number whose square is negative

No real number squares to a negative, so x2 = −1 has no real solution. Rather than stop there, mathematicians invented one: the imaginary unit i, defined by i2 = −1.

Attach it to the real numbers and you get the complex numbers, written a + bi, where a is the real part and b is the imaginary part. Far from being useless fictions, they turn out to describe rotations, waves, and electrical circuits.

\[ i^2 = -1, \qquad i^3 = -i, \qquad i^4 = 1 \]
Powers of i cycle every four: i, −1, −i, 1, then repeat. This four-step loop is a hint of rotation.
🔑 Two dimensions, not one
A real number sits on a line; a complex number needs a plane. Plot a + bi at the point (a, b): the horizontal axis is the real axis, the vertical axis is the imaginary axis. This picture is the Argand diagram, and it makes complex arithmetic visual.
🎮 The Complex Plane LIVE
Plot complex numbers on the Argand diagram; add and multiply them geometrically.

Adding is sliding

To add complex numbers, add the real parts and the imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i. On the Argand diagram this is exactly vector addition — each number is an arrow from the origin, and you add them tip-to-tail.

Multiplying is rotate-and-stretch

Multiply complex numbers by expanding like binomials and using i2 = −1. The geometric result is the beautiful part: multiplying by a complex number rotates and scales the plane. The distances from the origin (the moduli) multiply, and the angles from the positive real axis (the arguments) add.

\[ |z| = \sqrt{a^2 + b^2}, \qquad \arg(z) = \text{angle from the real axis} \]
The modulus |z| is the distance from the origin (by the Pythagorean theorem); the argument is the direction angle.
✨ Why i is a quarter-turn
The number i has modulus 1 and argument 90°. Multiplying by it adds 90° to any number's angle without changing its length — a quarter-turn counterclockwise. Do it four times and you have turned a full 360° back to where you started, which is exactly why i4 = 1.
📝 Worked example: Compute (3 + 2i) + (1 + 4i) and (2 + 3i)(1 − i).
  1. Add real and imaginary parts: \( (3+1) + (2+4)i = 4 + 6i \).
  2. Multiply by expanding: \( (2+3i)(1-i) = 2 - 2i + 3i - 3i^2 \).
  3. Use \( i^2 = -1 \): \( -3i^2 = +3 \), so it becomes \( 2 + 3 + ( -2 + 3)i = 5 + i \).
✓ The sum is <strong>4 + 6i</strong> and the product is <strong>5 + i</strong>.
📝 Worked example: Find the modulus of z = 3 + 4i.
  1. The modulus is the distance from the origin to the point \( (3, 4) \).
  2. By the Pythagorean theorem, \( |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} \).
  3. So \( |z| = 5 \).
✓ The modulus is <strong>|z| = 5</strong> — the number sits 5 units from the origin.
⚠️ i² = −1, so signs flip
When multiplying, do not forget that every i2 becomes −1. A term like −3i2 turns into +3, changing a real part. Skipping this substitution is the most common complex-multiplication mistake.

Check your understanding

1. What is i²?
By definition of the imaginary unit, i² = −1.
2. Where does the complex number 2 − 3i sit on the Argand diagram?
The real part 2 is the horizontal coordinate and the imaginary part −3 is the vertical, so it is 2 right and 3 down.
3. Compute (5 + 2i) + (1 − 6i).
Add parts separately: (5+1) + (2−6)i = 6 − 4i.
4. What is the modulus of 6 + 8i?
|z| = √(6² + 8²) = √(36 + 64) = √100 = 10.
5. Multiplying a complex number by i does what geometrically?
i has modulus 1 and argument 90°, so multiplying by it rotates a quarter-turn without changing length.
✅ Key takeaways
  • The imaginary unit i satisfies i² = −1, and its powers cycle i, −1, −i, 1.
  • A complex number a + bi plots at the point (a, b) on the Argand diagram (real axis horizontal, imaginary axis vertical).
  • Addition adds real and imaginary parts separately — vector addition on the plane.
  • Multiplication expands like binomials with i² = −1; geometrically, moduli multiply and arguments add.
  • The modulus |a + bi| = √(a² + b²) is the distance from the origin, and multiplying by i is a 90° rotation.