Complex Numbers

We've all heard of the term "roots", and we know that a root is the value(s) of x for which f(x) is 0. Great. But have you ever wondered why we need to find the roots of an equation? Such a simple, yet intriguing question!
To answer this, we need to go back to our definition of a root: a value of x for which f(x) is 0. This isn't the only way to look at it- we can also work out other definitions, like the one I'm about to tell you.

Let's say that I take two different functions, 2x² and  -7x - 5, and I want to find the values of x for which they're equal. We can write an equation that describes this:

2x² = -7x - 5. 

This can be rewritten as:

2x² + 7x + 5 = 0.

By writing the equation as a product of its factors: (x + 1)(2x + 5) = 0, we get the roots to be -5/2 and -1. 

So, if you have two real-valued functions f(x) and g(x), you can combine them and write an equation in the form  f(x) - g(x) = 0. So by solving for the equation (in other words, finding the roots of the equation), you're finding the value of x for which f(x) = g(x). Knowing when two functions are equal (which is the same as finding the intersection point of their graphs) is useful in lots of areas, like physics. Let's consider an example.

If an object is moving in one direction without a force acting on it, then it continues to move in that direction with a constant velocity. Let this velocity be v. If the object starts at point x = 0, the displacement of the object in time t will be x = vt. 

An object normally has forces acting on it. In the case of a car, it would be the friction in the brakes. We know that F = ma, where F is force, m is mass, and a is acceleration. If the force is constant, then acceleration is constant, since mass is always a constant.  

If the starting velocity is u, and v is the velocity in time t with constant acceleration a, then:

v = u + at

From the above expression, you can easily derive: 

 

This quadratic equation relates displacement (s) and time (t). It can tell you how much time it would take to travel a certain distance. A very important application is to find the stopping distance of a car travelling at a given velocity u. If you're driving your car and you apply the brakes, how long will it take to stop? If the car starts to travel with a constant deceleration -a (once you apply the brakes) and the velocity starts to decrease from u to 0, then by solving for t using the first equation and substituting in the second, we get the stopping distance s to be:

 

This shows that if you reduce your speed by, say, half, then your stopping distance becomes a fourth! Also, if you double your speed, then your stopping distance quadruples! This shows us how important reducing speed is, for your and others' safety. 

Now that we've looked at roots and their importance, let me ask you to find the roots of this equation:

x² - 1 = y.

The roots are obviously 1 and -1. You could find the roots this either by equating y to 0 and solving for x, or you could rewrite the equation as x² = 1 (so you're trying to find when two functions are equal), draw the graphs of f(x) = x² and g(x) = 1, and find the intersection points.

f(x) = x²

g(x) = 1

Obviously, the points of intersection are 1 and -1.

Now, find the roots of this equation:

x² + 1 = y

How can x² = -1? If you draw the graphs of f(x) = x² and g(x) = -1, you can see that there is no point of intersection. 

f(x) = x²

g(x) = -1

So, how can you solve the equation? Don’t worry, complex numbers are here to the rescue! Let's take If we define some number “i” such that i² = -1, we can see that x can be i or -i. Problem solved.

All complex numbers can be represented as z = a + bi, where a and b are real numbers and i is √-1. [Note: √-1 can also be denoted by j, but i is more frequently used.] The set of complex numbers is denoted by C. 

a is known as the real part of complex number z, and is denoted as a = Re(z). b is known as the imaginary part of complex number z, and is denoted as b = Im(z). So, you can see that real numbers are actually a subset of complex numbers!

R ∈ C

BASICS OF COMPLEX OPERATIONS:

• Addition/Subtraction:

(a + bi) ±(c + di) = (a + c) ± (b + di)

• Multiplication: (a + bi)(c + di)

= ac + (ad + bc)i + bd(i²) 

=  ac + (ad + bc)i - bd

• Division: (a + bi) ÷ (c + di) 

= [(a + bi)÷(c + di)][(c - di)÷(c - di)] 

= [ac + (bc - ad)i - bd] ÷ [c² + d²] 

POWERS OF i (apart from i⁰ = 1):

• i¹ = i
• i² = -1
• i³ = -i
• i⁴ = 1
• i⁵ = i  
• i⁶ = -1 …  

It is clear that the powers repeat every four times.

GRAPHICAL REPRESENTATION AND ARGUMENT OF A COMPLEX NUMBER: Complex numbers can be graphically represented on what’s known as the Argand plane. The Argand plane is similar to the XY-plane we’re all familiar with, but in the Argand plane the X-axis represents all real numbers and the Y-axis represents imaginary numbers (i, 2i, 3i…). 

The Argand Plane

In the above diagram, the magnitude, or length of z (denoted by |z|) is √(x² + y²). This can be proved using the Pythagoras theorem. Drop a perpendicular from the tip of z onto the real axis. The length of the horizontal arm of the right-angled triangle formed is obviously x, and the length of the vertical arm is y. On applying the Pythagoras theorem, you get the length of z (hypotenuse) to be √(x² + y²).  You could also drop the perpendicular onto the imaginary axis instead of the real axis- nothing changes.

The angle from the positive real axis to z is known as the argument of z, and is denoted by arg(z), or just Arg. The numeric value is given by the angle in radians and is positive if measured counterclockwise. 

In the above diagram, θ = arg(z). Since Arg is an angle, adding 2π will not change anything. So, Arg is what’s known as a multiple-valued function i.e., there are  multiple valid values of Arg (since θ is equivalent to θ + 2nπ for any integer n). To resolve this ambiguity, we assign what’s known as a principle value (denoted by pv). For example, in the square root (multiple-valued function), we take only the positive value by taking the principal root. 

The principal values for Arg are all values lying in:

• the open-closed interval (-π, π]. This means that all values from -π to π except for -π are included.
• the closed-open interval [0, 2π). This means that all values from 0 to 2π except for 2π are included.

Now, let us look at two of the most famous equations in complex numbers: Euler’s formula and de Moivre’s theorem. 

We’ll need to know a few important expansions (known as Maclaurin series, which are a type of what are known as Taylor series) to prove Euler’s formula, from which we can quickly derive de Moivre’s theorem. 

The expansions are: 

where x is the argument of a complex number z.

Euler’s formula: 

e^iθ = cosθ + isinθ

where θ is the argument of a complex number z.

Proof (I’ve used the variable x instead of θ): 

Substituting θ = π in Euler's formula gives us what is regarded as the most beautiful equation in math, dubbed as "our jewel" and "the most remarkable formula in mathematics" by Richard Feynman:

e^iπ + 1 = 0.

Now, let us look at de Moivre's theorem.

de Moivre’s theorem:

(cosθ + isinθ)ⁿ = cos(nθ)+ isin(nθ) 

Proof: 

e^iθ = cosθ + isinθ

From this, we can write:

(e^iθ)ⁿ = (cosθ + isinθ)ⁿ

We also know that:  

(e^iθ)ⁿ = (cosθ + isinθ)ⁿ = e^inθ

Using Euler’s formula, we get:

e^i(nθ) = cos(nθ)+ isin(nθ) 

Thus,

(cosθ + isinθ)ⁿ = cos(nθ)+ isin(nθ) 

POLAR FORMS OF COMPLEX NUMBERS:

In the above diagram, let z be the complex number represented, having argument θ. By dropping perpendiculars and forming right-angled triangles with both axes it can be shown that x = |z|cosθ and y = |z|sinθ, where |z| is the magnitude of z. So, we can substitute for x and y, giving the equation:

z = x + iy = |z|cosθ + i|z|sinθ = |z|(cosθ + isinθ)

Using Euler's formula, we get:

z = |z|e^iθ 

In other words:

z = |z|e^{i arg(z)}

This is known as the polar form of z. Using polar forms, we can look at something cool about multiplication by complex numbers- when you multiply a complex number by another, what happens is that the new complex number has an argument different from those of the original complex numbers. This is what I mean, mathematically:

Take the polar forms of any two complex numbers x and y: 

x = |x|e^iθ , y = |y|e^iω

where θ and ω are the arguments of complex numbers x and y respectively.

When you multiply the complex numbers with each other, you get:

xy = |x||y|(e^iθ)(e^iω)

Which is equivalent to:

|x||y|e^iθ + iω

Taking i common, we get:

xy = |x||y|e^i(θ + ω)

This means that the argument of the new complex number is θ + ω, which implies that there is rotation.

Multiplication by i results in rotation by 90 degrees. Here's what I mean: 

Here's a practical application of rotation. Let's say that you're travelling in your beat-up Toyota heading 3 units East for every 4 units North, and you need to make a 45 degree turn (in the anti-clockwise direction). How would you be heading after the turn?

It's pretty simple: your motion can be described as 3 + 4i. If you want to rotate by 45 degrees, you just need to multiply by 1 + i! 

(3 + 4i)(1 + i) = 3 - 4 + 7i = -1 + 7i

 So, after the turn you would be heading 1 unit West for every 7 units North.

This is my introduction to complex numbers, thanks for reading!