Imaginary Numbers and Me

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One of my favorite things in mathematics is imaginary numbers. Such a ridiculous name for something so important. I prefer the name complex numbers.

You probably haven’t seen an imaginary number since you were in school. The first time that you encounter  i is when learning to solve quadratic equations. You’ll find them in the quadratic formula when there is a negative number under the square root sign.  

Let a, b, and c be real number. The solutions of ax^2+bx+c=0 are 

    \[x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.\]

    \[i = \sqrt{-1}\]

So if we end up with a solution like

    \[x=2 +\sqrt{-2}\]

    \[x=2 +\sqrt{(-1)*2}\]

    \[x=2 +\sqrt{-1} *\sqrt{2}\]

We can re-write it as

    \[x=2+i\sqrt{2}\]

This is about as far as it goes for most people. I remember being delighted to learn about complex numbers. Then saddened that they disappeared from my life as soon as they had entered. I would see them again.

Rediscovering Complex Numbers

The only thing in physics that I liked was problems of electricity. To this day my favorite equations are Maxwell’s equations of electromagnetism. I am amazed that so few equations represent all you need to know. I was surprised to see that complex numbers make a reappearance when solving these types of problems. I just remember that being able to do this and breaking things down into component vectors allows us to solve them.

This is a terrible explanation. Unfortunately, I couldn’t find any basic videos for solving these types of problems. Kind of wished I had my old physics textbook.

The History of Imaginary Numbers

I don’t watch that many science channels on YouTube. I am happy to stick with the mathematics channels. But, YouTube has a way of recommending things so continuously that you have to check them out. I’ve had several old Veritasium videos in my feed lately. I finally watched this one on the history of imaginary numbers.

Video on Imaginary Numbers

Of course, they make the story more epic for views. See the thumbnail for what I mean. I dislike that to make math more interesting we have to make it more scandalous. I did enjoy the video though. Especially since it went over the connections between algebra and geometry.

It makes a connection to quantum mechanics and reality. Which, I think is a bit more spurious but I don’t know that much about those physics formulas. I’ve been watching rebuttals because of this. These sorts of big claims are interesting but I worry about people getting confused by them. If you don’t have a physics degree you don’t know if this claim is just clickbait or not.

One of the reasons why I don’t cover serious mathematics on this blog is because I don’t want to end up with this problem. I don’t want to make a claim that I can’t explain. That is why I’ll always defer to a source where a mathematician answers.

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