Welcome to the World of Complex Numbers!
In your math journey so far, you might have been told that you can't take the square root of a negative number. Well, in Further Pure Mathematics 1 (FP1), we unlock a whole new dimension where we can! Complex numbers are incredibly useful in engineering, physics, and advanced electronics. Don't worry if it seems strange at first; by the end of these notes, you'll be handling "imaginary" numbers like a pro.
1. What are Complex Numbers?
The foundation of this chapter is the number \( i \). We define \( i \) as the square root of \( -1 \).
Key Definition: \( i = \sqrt{-1} \), which means \( i^2 = -1 \).
The Standard Form: \( a + bi \)
A complex number \( z \) is usually written as:
\( z = a + bi \)
• \( a \) is the Real Part, written as \( \text{Re}(z) \).
• \( b \) is the Imaginary Part, written as \( \text{Im}(z) \).
Example: In \( z = 3 + 4i \), the real part is 3 and the imaginary part is 4.
Quick Review: Equality of Complex Numbers
Two complex numbers are equal only if their real parts are the same AND their imaginary parts are the same.
If \( a + bi = 5 - 2i \), then \( a = 5 \) and \( b = -2 \).
Did you know? Even though they are called "imaginary," these numbers are used to design airplane wings and understand how electricity flows through your home!
Takeaway: Every complex number has a real "anchor" and an imaginary "wing."
2. Basic Arithmetic with Complex Numbers
Working with complex numbers is very similar to basic algebra—just treat \( i \) like \( x \), but remember that \( i^2 \) always turns into \( -1 \).
Addition and Subtraction
Just add or subtract the "like" parts (reals with reals, imaginaries with imaginaries).
Example: \( (2 + 3i) + (4 - 5i) = (2+4) + (3-5)i = 6 - 2i \).
Multiplication
Use the FOIL method (First, Outer, Inner, Last), but be careful with the last term!
Step 1: Multiply \( (2 + 3i)(1 - 4i) \).
Step 2: \( 2(1) + 2(-4i) + 3i(1) + 3i(-4i) \).
Step 3: \( 2 - 8i + 3i - 12i^2 \).
Step 4: Since \( i^2 = -1 \), the last term becomes \( -12(-1) = +12 \).
Step 5: Simplify to \( 14 - 5i \).
The Complex Conjugate
If \( z = a + bi \), its conjugate (written as \( z^* \)) is \( a - bi \). You just flip the sign of the imaginary part.
Magic Trick: When you multiply a complex number by its conjugate, the result is always a real number!
\( (a + bi)(a - bi) = a^2 + b^2 \).
Division
To divide complex numbers, multiply the top and bottom by the conjugate of the bottom (the denominator).
Example: To solve \( \frac{1}{2+i} \), multiply the top and bottom by \( 2-i \).
Common Mistake: Forgetting to change the sign of the imaginary part when finding the conjugate. Remember: \( 3 - 4i \) becomes \( 3 + 4i \), but the real 3 stays positive!
Takeaway: Treat \( i \) like a variable, but always replace \( i^2 \) with \( -1 \).
3. The Argand Diagram
Complex numbers aren't just symbols; they are positions! An Argand Diagram is like a standard graph, but:
• The x-axis is the Real Axis.
• The y-axis is the Imaginary Axis.
A complex number \( z = x + iy \) is represented by the point \( (x, y) \).
Modulus and Argument
There is another way to describe a point's location: how far it is from the center and what angle it makes.
1. Modulus: The distance from the origin. It is written as \( |z| \).
Formula: \( |z| = \sqrt{x^2 + y^2} \) (Just like Pythagoras!)
2. Argument: The angle \( \theta \) measured from the positive real axis.
Formula: \( \tan \theta = \frac{y}{x} \).
Important: In FP1, we usually measure \( \theta \) in radians, where \( -\pi < \theta \leq \pi \).
Modulus Tip: The modulus is always a positive distance. If you get a negative number, check your square roots!
Takeaway: Argand diagrams turn algebra into geometry.
4. Modulus-Argument Form
Instead of writing \( z = a + bi \), we can use the modulus (\( r \)) and the argument (\( \theta \)):
\( z = r(\cos \theta + i\sin \theta) \)
To convert from \( a + bi \):
Step 1: Find \( r \) using \( \sqrt{a^2 + b^2} \).
Step 2: Find \( \theta \) using \( \tan^{-1}(\frac{b}{a}) \).
Step 3: Plug them into the formula.
Property to Remember:
\( |z_1z_2| = |z_1| \times |z_2| \)
The modulus of two numbers multiplied together is just the product of their individual moduli.
Takeaway: This form makes complex numbers look like they are dancing around a circle!
5. Solving Equations
This is where complex numbers really show their power. We can now solve quadratic equations that have no real roots.
Quadratic Equations
If you use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) and the part under the square root is negative, you use \( i \).
Example: \( x^2 + 9 = 0 \).
\( x^2 = -9 \)
\( x = \pm \sqrt{-9} = \pm 3i \).
The Conjugate Root Theorem
If an equation has real coefficients, complex roots always come in pairs! If \( a + bi \) is a root, then \( a - bi \) must also be a root.
• Quadratic equations: Have 2 roots (either 2 real OR 2 complex conjugates).
• Cubic equations: Have 3 roots (at least one must be real).
• Quartic equations: Have 4 roots (can be 4 real, 2 real and 2 complex, or 4 complex).
Step-by-Step: Solving a Cubic
If you are told \( 2 + i \) is a root of a cubic equation with integer coefficients:
1. Identify the second root: It must be the conjugate, \( 2 - i \).
2. Create a quadratic factor: Multiply \( (x - (2+i))(x - (2-i)) \).
3. Use long division: Divide the original cubic by this quadratic to find the final real root.
Don't worry if this seems tricky at first! Just remember: roots are like best friends; the complex ones never go anywhere without their conjugates.
Takeaway: Complex roots are predictable. Find one, and you've found two!
Summary Review Box
The Basics: \( i^2 = -1 \).
Conjugate: Flip the sign of the \( i \) part.
Division: Multiply by the conjugate of the bottom.
Modulus: Distance from origin \( \sqrt{a^2 + b^2} \).
Argument: Angle from the real axis.
Roots: Complex roots always appear in conjugate pairs (\( a \pm bi \)) for polynomials with real coefficients.