Welcome to the World of Complex Numbers!
In your previous math studies, you might have been told that you can't take the square root of a negative number. Well, in Further Maths, we break that rule! Complex Numbers allow us to solve equations that were previously "impossible." They are used everywhere in the real world, from designing airplane wings to understanding how electricity flows through your house.
Don't worry if this seems a bit "imaginary" at first—by the end of these notes, you'll see that these numbers follow very logical rules, just like the numbers you've used since primary school.
1. The Basics: What is \(i\)?
The foundation of this whole chapter is one simple definition: The imaginary unit, denoted by \(i\), is defined as:
\( i^2 = -1 \) (or \( i = \sqrt{-1} \))
Cartesian Form
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 \(Im(z)\).
Example: In \( z = 3 + 4i \), the real part is 3 and the imaginary part is 4.
The Complex Conjugate
If you have a complex number \( z = x + iy \), its conjugate (written as \(z^*\)) is simply the same number but with the sign of the imaginary part swapped: \( z^* = x - iy \).
Quick Review: A complex number is zero if and only if both its real and imaginary parts are zero.
Key Takeaway: Every complex number has a "real" side and an "imaginary" side. Think of it like a coordinate on a map!
2. The Argand Diagram
We can't fit complex numbers on a standard number line, so we use a 2D 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).
Analogy: If real numbers are like walking forward and backward on a tightrope, complex numbers are like having the whole floor to walk on!
Geometrical Effects
• Addition: Adding \(z_1 + z_2\) is just like adding vectors. You move along the real and imaginary components of both.
• Conjugate: Taking the conjugate \(z^*\) reflects the point across the Real Axis. It’s like looking at the number in a mirror placed on the floor.
3. Basic Arithmetic
Working with complex numbers is very similar to basic algebra—just remember the golden rule: \( i^2 = -1 \).
Addition and Subtraction
Just collect "like terms." Add the real parts together and the imaginary parts together.
Example: \( (2 + 3i) + (4 - i) = (2+4) + (3-1)i = 6 + 2i \).
Multiplication
Use the "FOIL" method (First, Outside, Inside, Last) or a grid to expand the brackets.
Crucial Step: Whenever you see \(i^2\), replace it with \(-1\).
Example: \( (2+i)(3+2i) = 6 + 4i + 3i + 2i^2 = 6 + 7i - 2 = 4 + 7i \).
Division
To divide complex numbers, we multiply the top and bottom by the conjugate of the denominator. This "realises" the denominator (makes it a real number).
Step 1: Find the conjugate of the bottom part.
Step 2: Multiply the top and bottom by this conjugate.
Step 3: Simplify using \(i^2 = -1\).
Key Takeaway: Treat \(i\) like a variable (like \(x\)), but always turn \(i^2\) into \(-1\).
4. Modulus and Argument
Instead of using coordinates (\(x\) and \(y\)), we can describe a complex number by its distance from the origin and its angle.
Modulus \( |z| \)
The modulus is the distance from the origin \((0,0)\) to the point \(z\). We use Pythagoras' Theorem:
\( |z| = r = \sqrt{x^2 + y^2} \)
Note: The modulus is always a positive number or zero.
Argument \( arg(z) \)
The argument (\(\theta\)) is the angle the line makes with the positive real axis, measured in radians.
• Angles measured anticlockwise are positive.
• Angles measured clockwise are negative.
The principal argument is usually taken in the interval \( (-\pi, \pi] \) or \( [0, 2\pi) \).
Did you know? In Further Maths, we almost always use radians instead of degrees. If your calculator says "D" at the top, switch it to "R"!
5. Modulus-Argument Form
We can express any complex number \(z\) as:
\( z = r(\cos\theta + i\sin\theta) \)
Where \(r\) is the modulus and \(\theta\) is the argument.
The Magic of Mod-Arg Arithmetic
Multiplying and dividing is much easier in this form!
• To multiply \(z_1\) and \(z_2\): Multiply the moduli and add the arguments.
\( |z_1 z_2| = r_1 r_2 \) and \( arg(z_1 z_2) = \theta_1 + \theta_2 \)
• To divide \(z_1\) by \(z_2\): Divide the moduli and subtract the arguments.
\( |\frac{z_1}{z_2}| = \frac{r_1}{r_2} \) and \( arg(\frac{z_1}{z_2}) = \theta_1 - \theta_2 \)
Quick Review Box:
To convert \( x+iy \) to \( r(\cos\theta + i\sin\theta) \):
1. \( r = \sqrt{x^2 + y^2} \)
2. \( \theta = \arctan(\frac{y}{x}) \) (but check the quadrant on your Argand diagram!)
6. Solving Equations
Complex numbers are the keys to unlocking polynomial equations.
Square Roots of a Complex Number
To find \(\sqrt{w}\), set \( (x+iy)^2 = w \). Expand the left side, compare the real and imaginary parts, and solve the resulting simultaneous equations.
Quadratic Equations
If you use the quadratic formula and get a negative number under the square root, you now know what to do! Just write it using \(i\).
Example: \( x^2 + 9 = 0 \rightarrow x^2 = -9 \rightarrow x = \pm 3i \).
The Conjugate Pair Theorem
For any polynomial equation with real coefficients, if a complex number \(z\) is a root, then its conjugate \(z^*\) is also a root.
• Cubic equations (power of 3) will have either 3 real roots OR 1 real root and 1 conjugate pair.
• Quartic equations (power of 4) will have 4 real roots, 2 real and 1 conjugate pair, OR 2 conjugate pairs.
Key Takeaway: Complex roots always travel in pairs (like shoes!). If you find \(2 + i\), you've automatically found \(2 - i\).
7. Loci on the Argand Diagram
A locus (plural: loci) is a set of points that satisfy a specific rule. In complex numbers, these rules create beautiful geometric shapes.
The Circle: \( |z - a| = k \)
This means "the distance between \(z\) and point \(a\) is always \(k\)."
• This represents a circle with centre \(a\) and radius \(k\).
Common Mistake: In \( |z + 2i| = 3 \), the centre is \(-2i\), not \(2i\). Always write it as \( |z - (-2i)| \).
The Perpendicular Bisector: \( |z - a| = |z - b| \)
This means "\(z\) is the same distance from \(a\) as it is from \(b\)."
• This represents the perpendicular bisector of the line segment joining points \(a\) and \(b\).
The Half-Line: \( arg(z - a) = \alpha \)
This means "the angle from point \(a\) to \(z\) is always \(\alpha\)."
• This represents a half-line (or ray) starting at \(a\) (but not including \(a\)) at an angle \(\alpha\). We usually draw an open circle at \(a\) to show it's excluded.
Regions and Inequalities
If the equation uses \( < \) or \( > \), it represents a region.
• \( |z - a| < k \) is the inside of a circle.
• Use solid lines for \(\le\) or \(\ge\) (meaning the boundary is included).
• Use dashed lines for \( < \) or \( > \) (meaning the boundary is excluded).
Quick Review: Set Notation
You might see a region written as \( \{z : |z - a| > k\} \). This just means "the set of all complex numbers \(z\) such that the distance from \(a\) is greater than \(k\)."
Summary Takeaway: Complex numbers aren't "imaginary" in the sense that they don't exist—they are just a more complete way of looking at the number system. Master the arithmetic, learn to "see" them on the Argand diagram, and you'll find they are one of the most powerful tools in your mathematical toolkit!