Welcome to the World of Complex Numbers!
In your GCSEs and A Level Maths, you might have 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 "unsolvable." They aren't just a mathematical trick; they are used in real-world engineering, physics, and even describing the way electricity flows. Don't worry if it seems strange at first—once you get the hang of the basic rules, it's just like algebra with a twist.
1. The Building Blocks: \( i \) and Cartesian Form
The core of this chapter is the imaginary unit, defined as: \( i = \sqrt{-1} \), which means \( i^2 = -1 \).
A complex number \( z \) is usually written in Cartesian form: \( z = x + iy \).
- \( x \) is the real part, written as \( Re(z) \).
- \( y \) is the imaginary part, written as \( Im(z) \). (Note: \( y \) itself is a real number!)
Arithmetic with Complex Numbers
Working with complex numbers is very similar to standard algebra—just treat \( i \) like a variable, but remember to replace \( i^2 \) with \(-1\).
- Addition/Subtraction: Add or subtract the real parts and the imaginary parts separately. Example: \( (2 + 3i) + (4 - i) = 6 + 2i \).
- Multiplication: Expand the brackets (FOIL). Example: \( (2 + i)(3 + 2i) = 6 + 4i + 3i + 2i^2 = 6 + 7i - 2 = 4 + 7i \).
- Division: To divide, you must multiply the top and bottom by the complex conjugate of the denominator. This "rationalises" the bottom.
The Complex Conjugate
If \( z = x + iy \), its complex conjugate is \( z^* = x - iy \).
When you multiply a number by its conjugate, the result is always a real number: \( z z^* = x^2 + y^2 \).
Quick Review: \( i^2 = -1 \). Treat \( i \) like a letter in algebra, but always simplify \( i^2 \) at the end!
2. Solving Polynomial Equations
A key skill in AQA Further Maths is solving equations where the roots might be complex.
Quadratic Equations
If you use the quadratic formula and the discriminant \( b^2 - 4ac \) is negative, you will get two complex roots. These will always be conjugate pairs (e.g., \( 2 + 3i \) and \( 2 - 3i \)).
Cubic and Quartic Equations
For polynomials with real coefficients:
- Cubic: Will have either 3 real roots OR 1 real root and 2 complex (conjugate) roots.
- Quartic: Will have 4 real roots, 2 real and 2 complex roots, or 4 complex roots (as two conjugate pairs).
Top Tip: If a question tells you that \( 1 + 2i \) is a root of a cubic equation with real coefficients, you automatically know that \( 1 - 2i \) is also a root!
3. The Argand Diagram
We can't fit complex numbers on a standard number line, so we use a 2D plane called an Argand Diagram.
- The x-axis is the Real axis.
- The y-axis is the Imaginary axis.
A number \( z = 3 + 2i \) is plotted as the point \( (3, 2) \).
Key Takeaway: An Argand diagram is just a way to "see" complex numbers as coordinates.
4. Modulus-Argument Form
Instead of \( x \) and \( y \), we can describe a complex number by its distance from the origin and the angle it makes with the positive real axis.
1. Modulus: The distance from the origin. \( r = |z| = \sqrt{x^2 + y^2} \).
2. Argument: The angle \( \theta \) measured from the positive real axis. \( \arg(z) = \theta \).
Important: Always use radians for the argument. The principal argument is usually given in the range \( -\pi < \theta \leq \pi \).
The Mod-Arg form is: \( z = r(\cos \theta + i \sin \theta) \).
Multiplying and Dividing in Mod-Arg Form
This is where Mod-Arg form is much faster than Cartesian form!
- To multiply: Multiply the moduli and add the arguments.
- To divide: Divide the moduli and subtract the arguments.
Did you know? This property comes from compound angle formulae in trigonometry!
5. Loci in the Argand Diagram
A "locus" is a set of points that satisfy a rule. You need to recognize these shapes:
- Circles: \( |z - a| = r \). This represents a circle with center at complex number \( a \) and radius \( r \).
- Half-lines: \( \arg(z - a) = \theta \). This is a ray starting at point \( a \) (but not including \( a \)) going off at angle \( \theta \).
- Perpendicular Bisectors: \( |z - a| = |z - b| \). This is the line exactly halfway between points \( a \) and \( b \).
Common Mistake: When finding the center of \( |z + 3 - 2i| = 5 \), remember to rewrite it as \( |z - (-3 + 2i)| = 5 \). The center is \( -3 + 2i \).
6. de Moivre’s Theorem and Exponential Form
de Moivre's Theorem
For any integer \( n \):
\( [r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta) \).
This is incredibly powerful for calculating large powers of complex numbers (like \( z^{10} \)) without expanding massive brackets.
Exponential Form (Euler's Relation)
We define \( e^{i\theta} = \cos \theta + i \sin \theta \).
This leads to the most compact way to write a complex number: \( z = re^{i\theta} \).
This form makes multiplication, division, and powers very easy using standard index laws.
Key Takeaway: \( z = x + iy = r(\cos \theta + i \sin \theta) = re^{i\theta} \). These are all the same number, just dressed in different clothes!
7. Roots of Complex Numbers
To find the \( n \)-th roots of a complex number (solving \( z^n = w \)):
1. Write the number \( w \) in mod-arg form.
2. Add multiples of \( 2\pi \) to the argument: \( \theta + 2k\pi \).
3. Use de Moivre’s theorem in reverse: \( z = w^{1/n} \).
4. This gives \( n \) distinct roots.
Geometric Fact: On an Argand diagram, the \( n \)-th roots of a number always form the vertices of a regular n-gon centered at the origin.
Roots of Unity
The roots of \( z^n = 1 \) are called the roots of unity. They always lie on a circle with radius 1. You can use these to solve complex geometric problems by treating shapes as points on the Argand diagram.
Summary Key Points:
- Use conjugates to divide Cartesian complex numbers.
- Complex roots of real-coefficient polynomials always come in pairs: \( a \pm bi \).
- Use Mod-Arg or Exponential form for powers and roots.
- Arguments must be in radians.
- The roots of \( z^n = w \) are spread evenly around a circle.