# Math Education Concepts

## COMPLEX NUMBERS …in brief

I don’t remember exactly when I learned about imaginary/complex numbers, but I do remember feeling excited about this “new” math; well, it was new to me.  I easily grasped the idea of complex numbers, but my classmates struggled with their meaning and purpose.  For me, it was easy; I treated i like a variable when performing operations on the complex numbers.  However, I understood very well the value of i as the square root of negative one.  I didn’t understand why it was so difficult for others to comprehend complex numbers.  Even now, some of my students do not “get it.”

When approaching the topic, I often ask my students to calculate the square root of negative one.  They look at me puzzled and say “one” or “negative one.”  I then ask “what is one squared” or “negative one squared” and they say “one.”  I repeat this cycle of questioning until they finally ask “what is the square root of negative one?”  Ah-ha!!  Now we’re getting somewhere.  This is usually how I introduce my students to complex numbers.  It never fails; my students almost always say “that’s stupid” or “what’s the point?”  After I allow them to vent their frustrations, I explain that i is short for “imaginary” (the name given by René Descartes in 1637) and represents the value of the square root of negative one.  I tell them the letter was assigned by a mathematician since the exact value is unknown; that i simply represents the value of the square root of negative one.  They buy it (well some of them), open their minds to it, and learn enough to get through their lessons.  In the meantime, I sigh with relief that they don’t ask additional questions.

For years, I’d wanted  research i and complex numbers so I could give a more accurate account of their beginnings, history, and existence.  Finally, the day came that I actually learned a brief history of these obscure numbers known as “complex numbers,” a term given by Carl Friedrich Gauss in 1831.  I’ll spare you the details.

As with many topics in mathematics, complex numbers became useful, and necessary.    Therefore, it was important for mathematicians to carefully analyze, define, introduce, and explain complex numbers.  This did not happen without opposition.  Some mathematicians argued against the usefulness of complex numbers.  Even Leonhard Euler did not care for the idea of these “impossible” numbers, but realized their necessity.  In spite of the arguments stacked against complex numbers, the quest to validate them did not end.  Many mathematicians have worked with complex numbers and found uses for them.

Delving any deeper into the world of complex numbers would require extensive research into other fields, which time does not allow.  Maybe one day I will have the resources to discover all the truly amazing characteristics of this number system.  As with many mathematical concepts, it took many centuries and multiple mathematicians to finally come up with yet another set of numbers.  I can’t help but wonder if there are any other number systems that have not yet surfaced.

NOTE:  Some of the information shared here was inspired by “Math through the Ages: A Gentle History for Teachers and Others,” written by William P. Berlinghoff and Fernando Q. Gouvêa.  A must read for Math Educators!