Welcome to the World of Complex Numbers!
In your math journey so far, you’ve been told that you can’t take the square root of a negative number. Well, in Further Mathematics, we break that rule! Complex numbers allow us to solve equations that were previously "impossible." Think of them not as "imaginary" in the sense of being fake, but as a new dimension of numbers that helps us understand everything from electrical engineering to quantum physics.
Don't worry if this seems a bit strange at first. We are just expanding our toolkit. If you can do basic algebra, you can do complex numbers!
1. The Basics: What is \(i\)?
The foundation of this chapter is a single definition: \(i^2 = -1\). This means that \(i = \sqrt{-1}\). We call \(i\) the imaginary unit.
A complex number is usually written in the form \(z = x + iy\), where:
- \(x\) is the real part, written as \(\text{Re}(z)\).
- \(y\) is the imaginary part, written as \(\text{Im}(z)\).
Arithmetic with Complex Numbers
Working with complex numbers is very similar to basic algebra (like collecting like terms with \(x\)).
- Addition/Subtraction: Just add or subtract the real parts and the imaginary parts separately.
Example: \((3 + 2i) + (5 - 4i) = (3+5) + (2-4)i = 8 - 2i\). - Multiplication: Expand the brackets as usual, but remember that \(i^2 = -1\).
Example: \((2 + i)(3 - i) = 6 - 2i + 3i - i^2\). Since \(-i^2 = -(-1) = 1\), the answer is \(7 + i\).
Quick Review: Whenever you see \(i^2\), immediately replace it with \(-1\). This is the most common place to lose marks!
Key Takeaway: Complex numbers have a real "half" and an imaginary "half." Treat \(i\) like a variable, but remember its special power to turn into \(-1\) when squared.
2. Complex Conjugates and Division
If you have a complex number \(z = x + iy\), its complex conjugate is written as \(z^* = x - iy\). You just flip the sign of the imaginary part.
Why is the conjugate useful?
When you multiply a number by its conjugate, the imaginary parts cancel out, leaving you with a purely real number:
\(z z^* = (x + iy)(x - iy) = x^2 + y^2\).
Division
To divide complex numbers, we use the conjugate to "realise the denominator" (similar to rationalising surds). Multiply the top and bottom by the conjugate of the bottom.
Example: To solve \(\frac{2+i}{1-i}\), multiply top and bottom by \((1+i)\).
Common Mistake: Forgetting to change the sign for the conjugate. If the bottom is \(3 + 4i\), the conjugate is \(3 - 4i\). If the bottom is \(3 - 4i\), the conjugate is \(3 + 4i\).
3. Solving Polynomial Equations
Now we can solve quadratic equations where the discriminant (\(b^2 - 4ac\)) is negative!
The Conjugate Pairs Rule
For any polynomial equation (quadratic, cubic, or quartic) with real coefficients, if a complex number \(z\) is a root, then its conjugate \(z^*\) must also be a root. They always travel in pairs!
Solving Cubics and Quartics
You might be given one root of a cubic equation, like \(z = 2 + i\). Because of the rule above, you immediately know another root is \(z = 2 - i\).
- Multiply the factors associated with these roots: \((z - (2+i))(z - (2-i))\) to get a quadratic factor.
- Use polynomial long division to divide the original equation by this quadratic factor to find the remaining roots.
Did you know? A cubic equation will always have at least one real root, because complex roots must come in pairs (2, 4, etc.), and a cubic has 3 roots total.
Key Takeaway: Roots of polynomials with real coefficients always come in conjugate pairs. If you find one complex root, you've actually found two!
4. The Argand Diagram
We can visualize complex numbers on an Argand Diagram. It looks just like a standard \(x\)-\(y\) graph, but:
- The horizontal axis is the Real axis.
- The vertical axis is the Imaginary axis.
Adding Geometrically: Adding two complex numbers on an Argand diagram is just like adding vectors. You can use the "parallelogram rule." The result is the diagonal of the parallelogram formed by the two numbers.
5. Modulus-Argument Form
Instead of using coordinates \((x, y)\), we can describe a complex number by its distance from the origin and its angle.
The Modulus \(|z|\)
The modulus is the distance from the origin to the point. We use Pythagoras:
\(|z| = r = \sqrt{x^2 + y^2}\).
The Argument \(\text{arg}(z)\)
The argument (\(\theta\)) is the angle the line makes with the positive real axis.
Crucial Rule: We always work in radians and usually keep the angle between \(-\pi\) and \(\pi\).
Mod-Arg Form: \(z = r(\cos\theta + i\sin\theta)\)
Helpful Trick: When finding the argument, always draw a quick sketch of the Argand diagram to see which quadrant the number is in. This stops you from getting the angle 180 degrees wrong!
Multiplying and Dividing in Mod-Arg Form
This is where the mod-arg form shines! It makes multiplication and division much easier:
- To multiply: Multiply the moduli and add the arguments.
\(|z_1 z_2| = |z_1||z_2|\) and \(\text{arg}(z_1 z_2) = \text{arg}(z_1) + \text{arg}(z_2)\). - To divide: Divide the moduli and subtract the arguments.
\(|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}\) and \(\text{arg}(\frac{z_1}{z_2}) = \text{arg}(z_1) - \text{arg}(z_2)\).
Key Takeaway: Mod-arg form turns multiplication into addition and division into subtraction. It's like magic for your calculator!
6. Loci and Regions
A locus (plural: loci) is a set of points that follow a specific rule. In the Argand diagram, these form shapes.
The Circle: \(|z - a| = r\)
This represents all points \(z\) whose distance from point \(a\) is exactly \(r\).
Interpretation: A circle with centre \(a\) and radius \(r\).
Example: \(|z - 3i| = 2\) is a circle centered at \((0, 3)\) with radius 2.
The Perpendicular Bisector: \(|z - a| = |z - b|\)
This represents points that are exactly halfway between point \(a\) and point \(b\).
Interpretation: A straight line that cuts the segment connecting \(a\) and \(b\) in half at a right angle.
The Half-Line: \(\text{arg}(z - a) = \theta\)
This represents all points starting from \(a\) that go off in the direction of angle \(\theta\).
Note: The point \(a\) itself is usually an open circle because the angle isn't defined exactly at the start point.
Regions
If you see inequalities like \(<\) or \(\leq\), you are shading an area.
- \(|z - a| \leq r\) means "inside or on the boundary of the circle."
- \(|z - a| > |z - b|\) means "the side of the line closer to \(b\)."
Quick Review Box:
1. \(|z - a| = r\) → Circle
2. \(|z - a| = |z - b|\) → Perpendicular Bisector
3. \(\text{arg}(z - a) = \theta\) → Half-line starting at \(a\)
Key Takeaway: Loci are just geometric rules. Translate the math into "distance from..." or "angle from..." and the shape will reveal itself!