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 
are
![\[x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:163/h:40/q:eco/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-58ec575013588a9879169fe7f44985f4_l3.png "Rendered by QuickLaTeX.com")
![\[x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:163/h:40/q:mauto/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-58ec575013588a9879169fe7f44985f4_l3.png "Rendered by QuickLaTeX.com")
![\[i = \sqrt{-1}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:68/h:18/q:eco/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-57461b879d1b6e0aae7c033d09f67249_l3.png "Rendered by QuickLaTeX.com")
![\[i = \sqrt{-1}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:68/h:18/q:mauto/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-57461b879d1b6e0aae7c033d09f67249_l3.png "Rendered by QuickLaTeX.com")
So if we end up with a solution like
![\[x=2 +\sqrt{-2}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:102/h:18/q:eco/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-330ebe56146ed42f3968d0340eb2a8fb_l3.png "Rendered by QuickLaTeX.com")
![\[x=2 +\sqrt{-2}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:102/h:18/q:mauto/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-330ebe56146ed42f3968d0340eb2a8fb_l3.png "Rendered by QuickLaTeX.com")
![\[x=2 +\sqrt{(-1)*2}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:145/h:22/q:eco/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-9a6a23b8c4c9689fc950f9c85bc24a2b_l3.png "Rendered by QuickLaTeX.com")
![\[x=2 +\sqrt{(-1)*2}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:145/h:22/q:mauto/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-9a6a23b8c4c9689fc950f9c85bc24a2b_l3.png "Rendered by QuickLaTeX.com")
![\[x=2 +\sqrt{-1} *\sqrt{2}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:142/h:19/q:eco/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-04e649bc1f9e6029cdfc8b5756f5541d_l3.png "Rendered by QuickLaTeX.com")
![\[x=2 +\sqrt{-1} *\sqrt{2}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:142/h:19/q:mauto/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-04e649bc1f9e6029cdfc8b5756f5541d_l3.png "Rendered by QuickLaTeX.com")
We can re-write it as
![\[x=2+i\sqrt{2}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:94/h:19/q:eco/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-c9bf9f2cc70f4d845c0885dd2b275754_l3.png "Rendered by QuickLaTeX.com")
![\[x=2+i\sqrt{2}\]](https://mlolsnzxbds5.i.optimole.com/tuRWnfw.flpX~b64b/w:94/h:19/q:mauto/https://beautyofmathematics.com/wp-content/ql-cache/quicklatex.com-c9bf9f2cc70f4d845c0885dd2b275754_l3.png "Rendered by QuickLaTeX.com")
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.
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.
I’m the writer with a mathematical muse. I love words, numbers, dreaming big & helping others. I believe that whatever you imagine, you can become. They/Them.
