I came to terms with the fact that I’m a geek a long time ago. Now I get to infect others.
“What?! What do you mean these numbers aren’t real?”
There is a long and sordid history of controversial numbers in mathematics. It took a long time for people to accept zero as a number. How can you define a number that’s essential nature is absence? They argued for centuries as to whether such a number could exist. Of course, now we recognize that zero is an important concept and a valid answer to equations.
One quick side note: as a teacher, I do wish that dividing by zero hurt. My life would be so much simpler.
Then there was the whole controversy about negative numbers. Mathematicians well into the 18th and 19th century were using them in their calculations, but often would refer to them as “absurd” or “fictitious” numbers. Many mathematicians refused to accept negative roots of their equations, insisting that they were false. The idea that a number could be less than nothing was pretty mind-boggling. We can conceptualize what 4 books look like, but what do -4 books look like? It is pretty hard to see how difficult this concept is now, if for no other reason that we have thermometers that read into negative numbers.
So it isn’t surprising to me that my students resist the notion of imaginary numbers. Taking the square root of negative numbers tends to break their fragile minds… and then we add insult to injury by saying that the combination of imaginary numbers and real numbers is a complex number. It almost seems redundant.
So here it is, the terrible truth that math teachers everywhere don’t tell their students: ALL numbers are imaginary. Every single number there is, every number that we can come up with, is a product of our minds. Can you actually pick up a 3? What does it look like?
I end up spending time trying to reassure my students that complex numbers really do have real-life applications, and weren’t just invented to torture unsuspecting minds. Impedance in circuitry. That the fundamental theorem of algebra states that all polynomials have roots in the set of complex numbers.
Here is the real reason why we teach imaginary numbers: they are cool. You can do things in complex numbers that are just plain magic.
I’m an evil math teacher… being enthusiastic about esoteric branches of magic math comes with the territory.