Welcome to Complex Numbers!
Hello there! If you thought mathematics only dealt with "real" numbers, get ready for an exciting journey into the world of Complex Numbers. This is where we break the rule that "you can't take the square root of a negative number," allowing us to solve many equations that were previously impossible.
Complex numbers are fundamental in advanced mathematics, engineering (especially electrical circuits and signal processing), and physics. Don't worry if this seems tricky at first—we will break down every concept step-by-step!
Section 1: The Imaginary Unit and Cartesian Form
1.1 Defining the Imaginary Unit (\(i\))
The foundation of complex numbers is the imaginary unit, denoted by \(i\).
We define \(i\) such that: $$i = \sqrt{-1}$$
This immediately leads to the most important property you must remember: $$i^2 = -1$$
Did you know? The term "imaginary" was coined historically because mathematicians were initially skeptical of these numbers, but they are just as mathematically valid and useful as real numbers!
1.2 The Cartesian Form of a Complex Number
A complex number, usually denoted by \(z\), is written in the Cartesian form (or rectangular form) as: $$z = x + iy$$
Here, \(x\) and \(y\) are both real numbers.
- \(x\) is the Real Part of \(z\), written \(Re(z)\).
- \(y\) is the Imaginary Part of \(z\), written \(Im(z)\). (Note: \(Im(z)\) is just \(y\), not \(iy\)).
Example: If \(z = 3 - 4i\), then \(Re(z) = 3\) and \(Im(z) = -4\).
Key Takeaway:
Complex numbers combine a real part and an imaginary part, allowing us to deal with \(\sqrt{\text{negative numbers}}\) via the key identity \(i^2 = -1\).
Section 2: Algebra of Complex Numbers (\(x + iy\))
Performing arithmetic with complex numbers is very similar to doing algebra with polynomials, treating \(i\) like a variable, but always remembering to simplify \(i^2\) to \(-1\).
2.1 Addition and Subtraction
To add or subtract complex numbers, you simply combine the real parts and combine the imaginary parts separately.
If \(z_1 = a + ib\) and \(z_2 = c + id\): $$z_1 + z_2 = (a+c) + i(b+d)$$
Analogy: Think of combining ingredients in a recipe. You combine the flour amounts together and the sugar amounts together; you don't mix flour and sugar totals!
2.2 Multiplication
Multiply complex numbers using the distributive law (just like FOIL for two brackets):
If \(z_1 = a + ib\) and \(z_2 = c + id\): $$z_1 z_2 = (a+ib)(c+id) = ac + iad + ibc + i^2bd$$
Now, remember that \(i^2 = -1\): $$z_1 z_2 = (ac - bd) + i(ad + bc)$$
Step-by-step example: Calculate \((2+3i)(1-i)\).
- Multiply out: \(2(1) + 2(-i) + 3i(1) + 3i(-i)\)
- Simplify: \(2 - 2i + 3i - 3i^2\)
- Replace \(i^2\) with \(-1\): \(2 + i - 3(-1)\)
- Combine: \(2 + i + 3 = 5 + i\)
2.3 The Complex Conjugate (\(z^*\))
The complex conjugate of \(z = x + iy\) is denoted \(z^*\) (or sometimes \(\bar{z}\)) and is found by changing the sign of the imaginary part: $$z^* = x - iy$$
Why is the conjugate important? When you multiply a complex number by its conjugate, the result is always a real number. $$z z^* = (x+iy)(x-iy) = x^2 - (iy)^2 = x^2 - i^2y^2 = x^2 + y^2$$
This is essential for the next operation: division.
2.4 Division (The Quotient)
To find the quotient of two complex numbers, \(\frac{z_1}{z_2}\), you must use the conjugate of the denominator, \(z_2^*\). This process is called realising the denominator.
Step-by-step example: Calculate \(\frac{2+i}{3-2i}\).
- Identify the denominator and its conjugate: Denominator is \(3-2i\), conjugate is \(3+2i\).
- Multiply the numerator and denominator by the conjugate: $$\frac{2+i}{3-2i} \times \frac{3+2i}{3+2i}$$
- Calculate the denominator (this should always be real): $$(3-2i)(3+2i) = 3^2 + 2^2 = 9 + 4 = 13$$
- Calculate the numerator: $$(2+i)(3+2i) = 6 + 4i + 3i + 2i^2 = 6 + 7i - 2 = 4 + 7i$$
- Write the final result in the form \(x+iy\): $$\frac{4+7i}{13} = \frac{4}{13} + i \frac{7}{13}$$
Quick Review: Algebra
- Addition/Subtraction: Group Real, Group Imaginary.
- Multiplication: FOIL and substitute \(i^2 = -1\).
- Division: Multiply numerator and denominator by the conjugate of the denominator.
Section 3: Roots and Solving Equations
3.1 Non-Real Roots of Quadratic Equations
When you solve a quadratic equation \(ax^2 + bx + c = 0\) where \(a, b, c\) are real coefficients, and you find that the discriminant (\(b^2-4ac\)) is negative, the roots will be complex numbers.
Crucially, these non-real roots always occur in conjugate pairs.
If \(z = \alpha + i\beta\) is a root of the polynomial, then \(z^* = \alpha - i\beta\) must also be a root.
Common Mistake to Avoid: This rule only applies if all the coefficients (\(a, b, c\)) of the polynomial are real!
3.2 Comparing Real and Imaginary Parts
A complex equation holds true only if the real parts on both sides are equal, AND the imaginary parts on both sides are equal.
If \(A + iB = C + iD\), then \(A=C\) and \(B=D\).
This technique is vital for solving equations involving complex numbers and their conjugates.
Example: Solving \(2z + z^* = 1 + i\)
Let \(z = x + iy\). Then \(z^* = x - iy\).
Substitute these into the equation:
$$2(x+iy) + (x-iy) = 1 + i$$
$$2x + 2iy + x - iy = 1 + i$$
$$3x + iy = 1 + i$$
Now, compare the parts:
Real parts: \(3x = 1 \implies x = \frac{1}{3}\)
Imaginary parts: \(y = 1\)
Therefore, the solution is \(z = \frac{1}{3} + i\).
Key Takeaway: Roots & Equations
Non-real roots of real-coefficient polynomials come in conjugate pairs. Solving complex equations requires equating the real parts and the imaginary parts separately.
Section 4: The Argand Diagram and Polar Form
4.1 The Argand Diagram
A complex number \(z = x + iy\) can be represented graphically as a point \((x, y)\) on a two-dimensional plane called the Argand diagram.
- The horizontal axis is the Real Axis.
- The vertical axis is the Imaginary Axis.
This geometric representation helps us understand the properties of complex numbers visually.
4.2 The Modulus (\(r\))
The modulus of a complex number \(z = x + iy\) is the distance of the point \((x, y)\) from the origin \((0, 0)\) on the Argand diagram. It is denoted by \(|z|\) or \(r\).
Using Pythagoras' theorem: $$|z| = r = \sqrt{x^2 + y^2}$$
The modulus is always non-negative. Note that \(|z|^2 = z z^*\).
4.3 The Argument (\(\theta\))
The argument of a complex number \(z\) is the angle \(\theta\) between the line segment connecting the origin to \(z\) and the positive Real axis. It is denoted \(\arg(z)\).
The argument is found using trigonometry: $$\tan \theta = \frac{y}{x}$$
We use the Principal Argument, which must lie in the range: $$-\pi < \theta \le \pi$$
Step-by-step guide to finding the Argument:
- Sketch: Draw the complex number on an Argand diagram to identify the quadrant.
- Reference Angle (\(\alpha\)): Calculate the acute angle \(\alpha = \tan^{-1}\left(\left|\frac{y}{x}\right|\right)\). Always use positive values for \(x\) and \(y\) here!
- Adjust: Find \(\theta\) based on the quadrant:
- Quadrant I (x>0, y>0): \(\theta = \alpha\)
- Quadrant II (x<0, y>0): \(\theta = \pi - \alpha\)
- Quadrant III (x<0, y<0): \(\theta = \alpha - \pi\) (or \(-\pi + \alpha\))
- Quadrant IV (x>0, y<0): \(\theta = -\alpha\)
Memory Aid: The principal argument ensures you always go the shortest angular distance from the positive real axis (either clockwise or anti-clockwise).
4.4 The Polar Coordinate Form
Using the modulus \(r\) and argument \(\theta\), we can express \(z\) in terms of polar coordinates.
From basic trigonometry: $$x = r \cos \theta \quad \text{and} \quad y = r \sin \theta$$
Substituting these into \(z = x + iy\) gives the Polar form: $$z = r(\cos \theta + i \sin \theta)$$
This form is often much easier for multiplication and division (though those techniques are covered in later modules, you must be able to convert between Cartesian and Polar forms now).
Key Takeaway: Argand & Polar Form
The Argand diagram maps \(z=x+iy\) to \((x, y)\). Modulus \(|z|\) is distance from origin; Argument \(\arg(z)\) is the angle from the positive real axis (in the range \((-\pi, \pi]\)).
Section 5: Simple Loci in the Complex Plane
A locus is a set of points that satisfy a geometric condition. In the complex plane, these conditions are usually defined in terms of modulus or argument.
5.1 Loci defined by Modulus (Distances)
5.1.1 Fixed Distance from a Point (Circle)
The locus of points \(z\) such that \(|z - a| = k\), where \(a\) is a fixed complex number and \(k\) is a fixed positive real constant.
This describes all points \(z\) whose distance from \(a\) is exactly \(k\).
Geometrical interpretation: A Circle with centre \(a\) and radius \(k\).
Example: \(|z - (2+i)| = 5\).
This is a circle centred at \((2, 1)\) with radius \(5\).
5.1.2 Equidistant from Two Points (Perpendicular Bisector)
The locus of points \(z\) such that \(|z - a| = |z - b|\), where \(a\) and \(b\) are fixed complex numbers.
This describes all points \(z\) whose distance from \(a\) is equal to their distance from \(b\).
Geometrical interpretation: The Perpendicular Bisector of the line segment joining \(a\) and \(b\).
Method: To find the Cartesian equation, substitute \(z=x+iy\), \(a=a_1+ia_2\), and \(b=b_1+ib_2\) into the equation and square both sides to eliminate the square roots.
5.2 Loci defined by Argument (Angles)
5.2.1 Fixed Argument from a Point (Half-Line)
The locus of points \(z\) such that \(\arg(z - a) = \theta\), where \(a\) is a fixed complex number and \(\theta\) is a fixed angle.
This describes all points \(z\) such that the vector connecting \(a\) to \(z\) makes an angle \(\theta\) with the positive Real axis.
Geometrical interpretation: A Half-Line (or ray) starting at \(a\), excluding the point \(a\) itself.
Example: \(\arg(z - 2) = \frac{\pi}{3}\).
This is a ray starting at \((2, 0)\) which extends outwards at an angle of \(60^\circ\) to the horizontal.
Key Takeaway: Loci
Modulus defines distance (circles or perpendicular bisectors). Argument defines angle (half-lines). Always identify the complex number being subtracted (the "start point" or "centre").