Introduction to Complex Numbers
Welcome to one of the most exciting chapters in Further Mathematics! Up until now, you've probably been told that you can't take the square root of a negative number. In Core Pure Mathematics, we throw that rule out the window. By introducing the imaginary unit \(i\), we unlock a whole new dimension of numbers. Don't worry if this seems a bit "unreal" at first—complex numbers are essential for engineering, physics, and even describing how light and sound behave!
1. The Basics: What is a Complex Number?
A complex number \(z\) is made of two parts: a real part and an imaginary part. We write it in Cartesian form:
\(z = x + iy\)
- \(x\) is the real part, denoted as \(Re(z)\).
- \(y\) is the imaginary part, denoted as \(Im(z)\).
- \(i\) is defined by the property \(i^2 = -1\).
Arithmetic with Complex Numbers
Working with complex numbers is very similar to basic algebra. Treat \(i\) like a variable (like \(x\)), but whenever you see \(i^2\), replace it with \(-1\).
- Addition/Subtraction: Add or subtract the real parts and imaginary parts separately.
Example: \((3 + 2i) + (1 - 4i) = (3+1) + (2-4)i = 4 - 2i\) - Multiplication: Use the FOIL method (First, Outside, Inside, Last).
Example: \((2 + i)(3 - i) = 6 - 2i + 3i - i^2 = 6 + i - (-1) = 7 + i\) - Division: To divide, multiply the top and bottom by the complex conjugate of the denominator to "realise" the bottom.
The Complex Conjugate
The complex conjugate of \(z = x + iy\) is written as \(z^* = x - iy\).
Memory Aid: Just "flip the sign" of the imaginary part!
Quick Review: \(z \times z^*\) will always result in a real number: \(x^2 + y^2\).
Key Takeaway
Complex numbers consist of a real and imaginary part. Treat \(i\) like algebra, but remember \(i^2 = -1\).2. Solving Polynomial Equations
In this course, you will solve quadratic, cubic, and quartic equations where the coefficients are real.
The Conjugate Pair Theorem
This is a vital rule: If a polynomial has real coefficients and a complex number \(z_1\) is a root, then its conjugate \(z_1^*\) must also be a root. Complex roots always come in pairs!
Step-by-Step: Solving Higher Degree Equations
If you are asked to solve a cubic equation \(f(z) = 0\):
- You will usually be given one root or a linear factor (like \(z+2\)).
- Use polynomial division or inspection to divide the cubic by the known factor.
- This leaves you with a quadratic equation.
- Solve the quadratic using the quadratic formula to find the remaining two roots (which might be complex).
Common Mistake: Forgetting that if \(3 + 2i\) is a root, \(3 - 2i\) is automatically a root. You don't need to be told both!
Key Takeaway
Complex roots of real-coefficient equations always appear in conjugate pairs.3. The Argand Diagram
Think of an Argand diagram as a map for complex numbers. Instead of an \(x\) and \(y\) axis, we have:
- The Real Axis (horizontal)
- The Imaginary Axis (vertical)
A complex number \(z = x + iy\) is simply a point \((x, y)\) or a vector from the origin to that point.
Modulus and Argument
We can also describe a complex number by how far it is from the origin and its angle.
- Modulus \(|z|\): The distance from the origin. Use Pythagoras: \(|z| = r = \sqrt{x^2 + y^2}\).
- Argument \(\arg(z)\): The angle \(\theta\) the vector makes with the positive real axis.
Measured in radians, where \(-\pi < \theta \leq \pi\).
Did you know? We use the "principal argument," which means we always take the shortest path from the positive real axis. If you are in the 2nd or 3rd quadrant, be careful with \(\arctan(y/x)\)—always draw a sketch to check your angle!
Key Takeaway
Argand diagrams turn numbers into geometry. Modulus is distance; Argument is direction.4. Modulus-Argument and Exponential Forms
Beyond the Cartesian form (\(x + iy\)), we have two other very useful ways to write complex numbers:
- Modulus-Argument Form: \(z = r(\cos\theta + i\sin\theta)\)
- Exponential (Euler) Form: \(z = re^{i\theta}\)
The Magic of Multiplication and Division
In these forms, math becomes much easier:
- To multiply: Multiply the moduli (\(r\)) and add the arguments (\(\theta\)).
- To divide: Divide the moduli (\(r\)) and subtract the arguments (\(\theta\)).
Quick Review Box:
\(\cos\theta = \frac{1}{2}(e^{i\theta} + e^{-i\theta})\)
\(\sin\theta = \frac{1}{2i}(e^{i\theta} - e^{-i\theta})\)
Key Takeaway
Use Cartesian (\(x+iy\)) for adding/subtracting. Use Mod-Arg or Exponential (\(re^{i\theta}\)) for multiplying/dividing.5. De Moivre’s Theorem
De Moivre’s Theorem is a superpower for dealing with powers of complex numbers. It states:
\([r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)\)
Basically, to raise a complex number to a power \(n\), you raise the modulus to that power and multiply the argument by \(n\).
Applications
- Multiple Angle Formulae: You can use De Moivre’s Theorem and Binomial Expansion to find expressions for \(\cos(n\theta)\) or \(\sin(n\theta)\) in terms of powers of \(\cos\theta\) and \(\sin\theta\).
- Series Summation: It can help sum geometric series that involve trigonometric terms.
Key Takeaway
De Moivre’s Theorem: Power the modulus, multiply the angle.6. Loci and Regions in the Argand Diagram
A locus (plural: loci) is a set of points that satisfy a specific rule. In complex numbers, these rules usually create circles or lines.
Common Loci to Recognise:
- Circles: \(|z - a| = b\). This is a circle with centre \(a\) and radius \(b\).
(Note: \(a\) is usually a complex number like \(2 + 3i\)). - Perpendicular Bisectors: \(|z - a| = |z - b|\). This is a straight line exactly halfway between points \(a\) and \(b\).
- Half-lines (Rays): \(\arg(z - a) = \theta\). This is a line starting at point \(a\) (but not including \(a\)) going off at angle \(\theta\).
Encouraging Phrase: Loci can feel abstract, but just think of them as "All points \(z\) that are exactly distance \(b\) away from \(a\)." It’s just a geometric description!
Key Takeaway
\(|z - a|\) is the distance between \(z\) and \(a\). Use this to "read" the geometric meaning of equations.7. Roots of Complex Numbers
Just as \(x^2 = 4\) has two roots (\(2\) and \(-2\)), a complex equation like \(z^n = re^{i\theta}\) has \(n\) distinct roots.
How to find the \(n\)-th roots:
- Write the number in exponential form: \(re^{i(\theta + 2k\pi)}\). We add \(2k\pi\) because sine and cosine repeat every full circle.
- Take the \(n\)-th root: \(z = r^{1/n} e^{i\left(\frac{\theta + 2k\pi}{n}\right)}\).
- Substitute \(k = 0, 1, 2, \dots, n-1\) to get your \(n\) different roots.
The Geometry of Roots
If you plot the \(n\)-th roots of a number on an Argand diagram, they will always form the vertices of a regular \(n\)-gon centered at the origin. For example, the 3 cube roots of a number will form an equilateral triangle!