Welcome to the World of Complex Numbers!
In your GCSE and standard A Level studies, you were taught that you couldn't find the square root of a negative number. Well, in Further Mathematics, we break those rules! Complex numbers allow us to solve "unsolvable" equations and are used in everything from electrical engineering to understanding how sound waves travel. Don't worry if it seems strange at first—once you get the hang of the basic "rules," it becomes one of the most logical parts of the course.
1. The Language of Complex Numbers
To start, we need a way to deal with \(\sqrt{-1}\). We define a new number, i, such that:
\(i^2 = -1\) or \(i = \sqrt{-1}\).
The Cartesian Form
A complex number \(z\) is usually written as:
\(z = x + iy\)
- x is the real part, written as Re(z).
- y is the imaginary part, written as Im(z).
The Complex Conjugate
If \(z = x + iy\), its conjugate, written as \(z^*\), is simply:
\(z^* = x - iy\)
Think of it like a reflection in a mirror—you just flip the sign of the imaginary part.
Quick Review:
- \(i^2 = -1\)
- \(z = \text{Real} + i(\text{Imaginary})\)
- To find the conjugate, change the sign of the \(i\) term.
Key Takeaway: Complex numbers have two parts—a real part and an imaginary part. They always function as a pair!
2. The Argand Diagram
We visualize complex numbers on a graph called an Argand Diagram.
- The horizontal axis is the real axis (like the x-axis).
- The vertical axis is the imaginary axis (like the y-axis).
Visualizing Operations:
- Addition/Subtraction: This works exactly like vector addition. If you add two complex numbers, you are effectively joining two arrows end-to-end.
- Conjugates: On an Argand diagram, \(z^*\) is a reflection of \(z\) across the real axis.
Did you know? Argand diagrams are named after Jean-Robert Argand, an amateur mathematician who worked as a bookkeeper in Paris!
3. Modulus and Argument
Instead of using coordinates \((x, y)\), we can describe a complex number by its distance from the center and its angle. This is the modulus-argument form.
The Modulus \(|z|\)
The modulus is the distance from the origin \((0,0)\) to the point. We use Pythagoras' Theorem:
\(|z| = r = \sqrt{x^2 + y^2}\)
The Argument \(arg(z)\)
The argument is the angle \(\theta\) that the line makes with the positive real axis.
- It is measured in radians.
- Principal Argument: To keep things unique, we usually give the angle in the range \(-\pi < \theta \leq \pi\) or \(0 \leq \theta < 2\pi\).
- \(\tan \theta = \frac{y}{x}\)
Common Mistake: Always draw a quick sketch to check which quadrant your number is in! If \(z = -1 - i\), the calculator might give you an angle in the wrong direction. A sketch ensures your angle points the right way.
Key Takeaway: Modulus = Distance; Argument = Angle.
4. Different Forms of Complex Numbers
You need to be comfortable switching between these three ways of writing the same number:
- Cartesian: \(z = x + iy\) (Great for adding and subtracting).
- Modulus-Argument: \(z = r(\cos \theta + i\sin \theta)\) (Often shortened to \(r\text{cis}\theta\) or \([r, \theta]\)).
- Exponential Form: \(z = re^{i\theta}\) (The most powerful form for multiplication and powers).
Converting: Step-by-Step
- Find \(r\) using \(\sqrt{x^2 + y^2}\).
- Find \(\theta\) using \(\tan^{-1}(\frac{y}{x})\) and checking the quadrant on a sketch.
- Plug them into the formula \(r(\cos \theta + i\sin \theta)\) or \(re^{i\theta}\).
5. Basic Operations
Addition and Subtraction
Use Cartesian form. Simply add the real parts together and add the imaginary parts together.
Example: \((3 + 2i) + (1 + 4i) = 4 + 6i\).
Multiplication and Division
You can do this in Cartesian form (remembering \(i^2 = -1\)), but it’s much faster in Mod-Arg or Exponential form!
Multiplication Rule: Multiply the moduli, add the arguments.
\(z_1 z_2 = [r_1 r_2, \theta_1 + \theta_2]\)
Division Rule: Divide the moduli, subtract the arguments.
\(\frac{z_1}{z_2} = [\frac{r_1}{r_2}, \theta_1 - \theta_2]\)
Key Takeaway: Multiplication stretches/shrinks the number and rotates it!
6. Solving Equations
The Conjugate Pair Theorem
If a polynomial equation (like a quadratic or cubic) has real coefficients, then any complex roots must come in conjugate pairs.
If \(2 + 3i\) is a root, then \(2 - 3i\) must also be a root.
Finding Square Roots
To find the square root of \(a + ib\), let \((x + iy)^2 = a + ib\).
1. Expand the left: \(x^2 - y^2 + 2xyi = a + ib\).
2. Compare the real parts: \(x^2 - y^2 = a\).
3. Compare the imaginary parts: \(2xy = b\).
4. Solve these simultaneous equations to find \(x\) and \(y\).
7. Loci in the Argand Diagram
A "locus" is a set of points that follow a specific rule. You need to recognize these four types:
- The Circle: \(|z - a| = k\)
This means "the distance from point \(a\) to \(z\) is always \(k\)". It's a circle with center \(a\) and radius \(k\). - The Perpendicular Bisector: \(|z - a| = |z - b|\)
This means "\(z\) is the same distance from \(a\) as it is from \(b\)". It is the straight line exactly halfway between \(a\) and \(b\). - The Half-line: \(arg(z - a) = \theta\)
This is a ray starting at point \(a\) and heading off at an angle \(\theta\). Note: the point \(a\) itself is usually excluded (shown with an open circle). - Vertical/Horizontal Lines: \(Re(z) = k\) or \(Im(z) = k\).
Encouraging Phrase: If a question uses an inequality like \(|z - a| \leq k\), you just shade the inside of the circle. Solid lines are for \(\leq\) or \(\geq\), and dashed lines are for \(<\) or \(>\).
8. De Moivre’s Theorem
This theorem is a lifesaver for powers:
\((r(\cos \theta + i\sin \theta))^n = r^n(\cos n\theta + i\sin n\theta)\)
In simple terms: To raise a complex number to a power \(n\), you raise the modulus to the power \(n\) and multiply the argument by \(n\).
Applications
- Multiple Angle Formulae: You can use De Moivre’s to express things like \(\cos 3\theta\) in terms of \(\cos \theta\).
- Finding nth Roots: To find the roots of \(z^n = w\), remember that there will be \(n\) roots, and they will form the vertices of a regular n-gon (like a square for 4 roots, or a pentagon for 5 roots) centered at the origin.
Roots of Unity
The "roots of unity" are the solutions to \(z^n = 1\). They always lie on a circle with radius 1 and are spaced perfectly evenly around the center.
Summary Key Takeaways:
1. Use \(i^2 = -1\).
2. Cartesian for add/subtract; Exponential for multiply/divide/powers.
3. Always sketch your Argand diagram.
4. Roots of polynomials with real coefficients always come in pairs (\(z\) and \(z^*\)).
5. De Moivre's turns powers into simple multiplication of the angle.