Math, the Way it Should Be (Unit 3: Imaginary Numbers, Revisited)

It’s been a long time since the last math lesson.  I have been busy with other interests of mine, though I definitely haven’t forgotten about this whole math thing.  If you’re not sure what I’m talking about you can visit the first ever math post, the first post in the first unit, the second unit, or the last math post.  I have decided to do a minor detour in the curriculum to revisit imaginary numbers which I kind of brushed over in a previous lesson.  I do recommend reviewing that lesson if you’re unfamiliar with imaginary numbers.  I didn’t include this exploration of this topic in another unit because it doesn’t really fit in with the areas of mathematics that we’ll be headed into later (until you get into the really weird stuff) so we probably won’t be seeing them for a long, long while, if at all.  And since I can’t really cover this entire topic in one lesson and don’t want to spend more than that, I’ll just hit some of the highlights and things that I think are interesting about them.

One thing you’ll want to know about imaginary and complex numbers is about their modulus.  The modulus is akin to the absolute value of a real number.  Essentially, it gives you the distance the number is from 0.  That is, 4 is 4 away from zero, so $|4| = 4$.  In the same way, -4 is 4 away from zero, so $|-4| = 4$. If you think of a complex number as two separate vectors on the complex plane, you can think of the modulus as the length of the vector that they create when added together.  In other words, you use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle.  For example, if you have the complex number \(z = ai + b\) and want to find it’s modulus, you would take the real part ($b$) and the imaginary part (\(a\)), square them, multiply them together, and then take the square root.  This means that

$|z| = \sqrt{ a^2 + b^2 }$.

The modulus of complex numbers has an interesting property.  If you think about a triangle with two sides made up of the vectors representing complex numbers in the complex plane (visual coming to help sort that out!), you will be able to see that the length of the third side cannot be greater than the sum of the length of the other two sides!  Visually, this looks like this:

Mathematically, it is known as the triangle equality and is written like this:

$|x| +|y| \ge |x + y|$

This is a helpful tool when you’re looking to prove certain things about complex numbers.

Another interesting feature of complex numbers is the ability to use them to transform exponential expressions (which will be showing up often, later) into expressions involving trigonometric functions.  This is for reasons that I do not want to prove here, but let me throw some math down on the page:

\(e^{ai} = cos(a) + i*sin(a)\)

Here it is good to do a “sanity check” and substitute 0 into the equation for \(a\).  We know that any number raised to the 0 power is equal to 1.  We also know that \(cos(0) = 1\) and \(sin(0) = 0\) so this equation is consistent.  If you substitute \(\pi\) into the equation for \(a\), you will find a very important result:

\(e^{i\pi} + 1 = 0\)

This so-called Euler Equation relates pretty much all of the most important mathematical numbers.  It’s pretty elegant and that’s probably why people like it so much.

Homework:

Take a break.  I have assigned far less homework than I originally expected but that’s okay.  I’m not collecting it and in all likelihood, you’re not doing it.  I don’t know how many math posts I’ll be doing in September (if I had to guess, I’d say none) because of a challenge I’ll be undertaking (more on that tomorrow!).

-Lane