Top 5 facts about Imaginary Math (Synopsis)

There is not enough love and goodness in the world to permit giving any of it away to imaginary beings.” -Friedrich Nietzsche

You know that the square root of -1 is “i”, an imaginary number. But did you know any of these?

For classical (Newtonian) mechanics, as well as electromagnetism, it’s sufficient to use real mathematics to describe it fully, if not always ideal. But to take care of Einstein’s relativity (and spacetime in particular), as well as quantum mechanics, you need to go beyond the real numbers and into the complex plane, which includes imaginary numbers as well!

Image credit: Sven Geier of

Image credit: Sven Geier of

From the square root of i to i^i power, here are my Top 5 facts about Imaginary Numbers! If yours are different, weigh in below!


  1. #1 Alan Armstrong
    February 26, 2014

    First and foremost. Exclamation points don’t make the article more exciting! STOP IT.
    The article has its points, but the format needs some cleaning up. It’s not a print magazine – stop trying to format as one (i.e. why are you putting a half-page header image on something people may be viewing on a phone? All you’re doing is pissing me off. Second, cool images are nice, and credits are important – but *captioning* the images is just as important. You had one last week where the images provided no connection to the article and I couldn’t connect the text to the embeds. Zoom on images is also useful considering the various devices that may view the content.

  2. #2 Nova
    February 26, 2014

    In the first “fact”, where you square the both sides … i think it should be x(2) + 2xyi +y(2)i(2), thus making the real aprt equal to x(2) – y(2). 🙂

  3. #3 Ethan
    February 26, 2014

    Thank you, Nova; when I do a long and technical post, I always know there’s likely to be a careless typo somewhere. Thanks for finding it. I fixed it, and let’s hope it’s the only one!

    Alan, thanks for the constructive criticism. Let’s see if I can incorporate any of that into future posts.

  4. #4 darkgently
    February 27, 2014

    Nice post Ethan. Thanks for keeping the comments going here too.

    Your #5 fact is actually more amazing than you realise. Complex exponents are multi-valued (because the complex logarithm is) so there are actually infinitely many possible values for i^i… *all* of which are real! 🙂

  5. #5 Sinisa Lazarek
    February 27, 2014

    Very interesting post. Being very low skilled in math, something that even I can admire is great 🙂

    I was blown away by the relation e^iPi + 1 = 0

    when you think about.. they are just numbers.. abstracts.. yet these 3 are almost in every physical law.. thus Universe.. and yet.. what a relation… so elegant… wow!

  6. #6 dean
    February 27, 2014

    Very nice to see this (in spite of the foolish complaints in the first post).
    It’s amazing how pushback against areas of math exist among certain groups – look at conservapedia’s dismissal of complex numbers and non-belief in the validity of proof by contradiction, as two examples.
    On an almost unbelievable note, Arizona State Senator Al Melvin (who was a backer of the now vetoed bill that would have made discrimination legal) opposes the Common Core standards. There may be legitimate concerns over these standards, but one of his reasons for opposing them is, apparently basic algebra:

    Pressed by [Sen. David] Bradley for specifics, Melvin said he understands “some of the reading material is borderline pornographic.” And he said the program uses “fuzzy math,” substituting letters for numbers in some examples.

  7. #7 tcmJOE
    February 27, 2014

    One of the most interesting bits for me is that once you have complex numbers, you can solve very algebraic equation. Once you’ve got the integers and try and solve for x, you’ll quickly run into the need for complex numbers. But you don’t need to go any farther (no need to let the quaternions or octonions out of the closet).

    Also, more beautiful than e^{i \pi} + 1 = 0? e^{i \tau} = 1 + 0. I fully support the use of {\tau} = 2{\pi} whenever possible–it makes dealing with phase angles (and Fourier analysis, and probability arguments, and n-ball volumes, and…) so much eaiser. Embrace the tau.

  8. #8 G
    February 27, 2014

    While we have a bunch of mathematicians here, what do y’all think of Max Tegmark’s multiverse theories?

    It seems to me he’s ultimately arguing for something like a kind of Platonic reality where the universe isn’t just _described_ by math, but at its most fundamental level, _is_ math. About which I’m inclined toward (open-minded) skepticism, but I’d like to hear from some people who are in a better position to evaluate his ideas.

  9. #9 brian
    February 27, 2014

    forget what debbie downer said up there. i enjoy the enthusiastic exclamation points !!!!!!!!

  10. #10 Wow
    February 28, 2014

    G, math’s application to reality may be more about us being able to do so, therefore picking out those patterns of reality that can be done with maths.

    Although you can use Shroedinger’s equation to work out how the cat exists, it won’t help much in working out why it likes milk.

  11. #11 eric
    March 4, 2014

    i always wierded me out, until someone described the geometric interpretation of it. Plot real numbers on the x-axis and imaginary ones on the y. What would the function “multiply by -1” be? A 180 degree rotation. So then what is the function “multiply by i?” A 90-degree rotation. Do it twice (X*i*i) and you get -X. Simple.

  12. #12 Sai
    March 8, 2014

    I have worked with complex numbers before but never realized how -1 = 1 (incorrectly of course!)

    I think that the reason why the multiplication of the square roots is wrong is that there may be multiple square roots (as opposed to a single value for a^m where m>0) and combining the square roots that way may be equivalent to multiplying different roots together. E.g. -1 = i*(-i)
    Is this correct?

    Thank you for the great blog!

  13. #13 Bill Seymour
    St. Louis, MO
    March 14, 2014

    OK, let’s see how much HTML I can use in comments:


    (I’ll see whether that works when I submit the comment. In any event, it’ll probably be close enough.)

    But the point of my comment is not that experiment. Rather, I want to assert that the beauty of that equation is an esthetic reason to not replace π with τ. 😎

    [[[Ethan’s note: I fixed your html.]]]

  14. #14 Kent Swearingen
    Tulsa, OK
    June 26, 2014

    In your second fact you repeatedly use the word “unique”, which means “the only one” in mathematics, although it can have other meanings in ordinary language. The point you are making is that the solutions are in fact different, and the word you want for this is”distinct”, not “unique”.

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