Welcome to the World of Complex Numbers!
In your GCSEs and earlier A Level work, you were probably told that you cannot find the square root of a negative number. While that is true for "Real" numbers, mathematicians found that by inventing a new number called \( i \) (where \( i^2 = -1 \)), a whole new dimension of mathematics opens up! Complex numbers are used everywhere from designing airplane wings to understanding how electricity flows through your home. Don't worry if it seems a bit strange at first—it’s just like moving from a 1D line to a 2D map.
1. What is a Complex Number?
A complex number is made of two parts and is usually written in the form \( z = x + iy \).
- \( x \) is called the real part.
- \( y \) is called the imaginary part.
Example: In the complex number \( z = 3 + 4i \), the real part is 3 and the imaginary part is 4.
Quick Review: The Power of \( i \)
The most important thing to remember is that \( i^2 = -1 \). Whenever you see an \( i^2 \) in your working, replace it with -1 immediately!
Key Takeaway: Every complex number has a real part and an imaginary part. They are like coordinates on a map, which we will see later in Argand diagrams.
2. Adding, Subtracting, and Multiplying
Good news! Adding and subtracting complex numbers is just like "collecting like terms" in algebra.
Addition and Subtraction
To add or subtract, you simply combine the real parts together and the imaginary parts together.
Example: \( (2 + 3i) + (4 - i) = (2 + 4) + (3i - i) = 6 + 2i \)
Multiplication
To multiply, you use the FOIL method (First, Outside, Inside, Last) just like you do with brackets in GCSE. The only trick is to remember that \( i^2 = -1 \).
Step-by-step example: \( (2 + 3i)(1 + 4i) \)
1. First: \( 2 \times 1 = 2 \)
2. Outside: \( 2 \times 4i = 8i \)
3. Inside: \( 3i \times 1 = 3i \)
4. Last: \( 3i \times 4i = 12i^2 \)
5. Combine: \( 2 + 11i + 12(-1) = 2 + 11i - 12 \)
6. Final Answer: \( -10 + 11i \)
Key Takeaway: Treat \( i \) like \( x \) in algebra, but always turn \( i^2 \) into -1.
3. The Complex Conjugate and Division
The complex conjugate of a number \( z = x + iy \) is written as \( z^* = x - iy \). You just flip the sign of the imaginary part.
Why is it useful?
When you multiply a complex number by its conjugate, the imaginary parts disappear, and you get a real number: \( (x+iy)(x-iy) = x^2 + y^2 \).
Division
To divide complex numbers, we use a trick: multiply the top and bottom by the conjugate of the denominator (the bottom part).
Analogy: This is exactly like "rationalising the denominator" when you worked with surds (roots) in Core Maths!
Key Takeaway: Use the conjugate to "clean up" the bottom of a fraction so it's no longer imaginary.
4. Solving Equations
In Further Maths, you will solve equations that have no real solutions.
Quadratic Equations
If you use the quadratic formula and get a negative number under the square root (the discriminant), don't stop! Just use \( i \).
Example: If you get \( \sqrt{-16} \), write it as \( 4i \).
Cubic and Quartic Equations
For equations with real coefficients (the numbers in the equation), there is a golden rule: Complex roots always come in conjugate pairs.
If you are told that \( 2 + 3i \) is a root of an equation, you automatically know that \( 2 - 3i \) is also a root!
Common Mistake: Students often forget that for a cubic (degree 3) or quartic (degree 4) equation, you might have some real roots and some complex roots. Use the pair rule to find the complex ones first.
Key Takeaway: Roots are like best friends; if a complex root shows up, its conjugate pair is always there too.
5. Argand Diagrams
An Argand diagram is just a graph where we plot complex numbers. The x-axis is the Real axis and the y-axis is the Imaginary axis.
To plot \( z = 3 + 2i \), you simply go 3 units right and 2 units up. It’s that simple!
Key Takeaway: Argand diagrams turn abstract numbers into visual points (or vectors) on a plane.
6. Modulus and Argument
Instead of using \( x + iy \) (Cartesian form), we can describe a point by how far it is from the center and what angle it makes.
- Modulus \( |z| \): The distance from the origin to the point. Use Pythagoras: \( |z| = \sqrt{x^2 + y^2} \).
- Argument \( \arg(z) \): The angle measured from the positive real axis. Usually measured in radians.
The Mod-Arg Form
You can write any complex number as: \( z = r(\cos \theta + i\sin \theta) \)
Where \( r \) is the modulus and \( \theta \) is the argument.
Memory Aid: SOH CAH TOA
Since this forms a right-angled triangle on the Argand diagram, you can use basic trig. Just be careful which quadrant your point is in when calculating the angle!
Multiplying and Dividing in Mod-Arg Form
This is where Mod-Arg form becomes a superpower. It makes multiplication and division much easier:
- To multiply: Multiply the moduli (\( r \)), but add the arguments (\( \theta \)).
- To divide: Divide the moduli (\( r \)), but subtract the arguments (\( \theta \)).
Key Takeaway: Modulus is "how far," and Argument is "which way."
7. Loci in the Argand Diagram
A locus (plural: loci) is a set of points that follow a specific rule. At AS Level, you need to know two main types:
1. The Circle: \( |z - a| = r \)
This represents all points \( z \) that are a fixed distance \( r \) away from a point \( a \).
Analogy: Imagine a dog on a lead of length \( r \) tied to a post at position \( a \). The dog can walk anywhere to form a circle.
Note: If the sign is \( |z - a| > r \), it means everything outside the circle.
2. The Half-Line: \( \arg(z - a) = \theta \)
This is a line starting at point \( a \) and heading off at an angle \( \theta \).
Analogy: A spotlight placed at point \( a \) pointing in a specific direction.
Quick Review Box:
- \( |z - (3+2i)| = 5 \) is a circle centered at (3, 2) with radius 5.
- \( \arg(z - i) = \frac{\pi}{4} \) is a line starting at (0, 1) going up at 45 degrees.
Key Takeaway: Loci equations are just "rules" that draw shapes on your Argand diagram. Always identify the "starting point" (the \( a \) value) first!
Congratulations!
You've covered the core essentials of Complex Numbers for AQA AS Further Maths. Remember, the key is to keep your Real and Imaginary parts separate, watch your \( i^2 \) signs, and use Argand diagrams to help you visualize what's happening. You've got this!