Imagine All the Numbers…



A fair question: how did “i” get the name of “imaginary number”?

It seems harsh. In some sense, all numbers are imaginary. After all, is there really such a thing as negative numbers? You can’t have -2 friends, no matter how alienating your Facebook posts are.


Or what about the irrationals? If you take a 1-meter stick and mark it up into equal segments, then no matter how tiny and minute the divisions, you’ll never get an irrational length. Even if you go down to the atomic level. That’s kind of weird.

Heck, what about the natural numbers, like 7 and 15? Isn’t it a little weird to pretend that these exist? I mean, 7 what? Numbers are made for counting. How can you have a number without anything to enumerate?


So sure, I’ll grant that is imaginary, but only insofar as every number is!

Of course, back in the day, mathematicians saw something fishy about these numbers. After all, they’re neither bigger than zero, nor smaller than zero, nor equal to zero. THey give negative results when squared. So you can’t blame mathematicians like Euler for using names like these:

  • “imaginary”
  • “impossible”
  • “inconceivable”
  • “fancied”

Clearly, we ought to envy the inhabitants of the nearby parallel universe where these are called “fancied” numbers. But I, for one, pity those poor souls in the universe where they are called “inconceivable,” which lacks the playful color of “imaginary.”

(The word “imaginary,” by the way, came from Descartes. Like many other great names, it began as a slur.)



It’s fun to try to think of other names for i and its multiples.

“Orthonumbers” or “orthogonal numbers” seems like a popular choice. It was the first that came to my mind, and I’m not alone. After all, they appear not on the number line, but perpendicular (or “orthogonal”) to it.

I also think we could call real numbers “posroots” (since they are the square roots of positive numbers) and imaginary numbers “negroots” (since they are the square roots of negative numbers).

(Bonus for “Guardians of the Galaxy” fans: you get to picture i saying “I am negroot.”)

Do names really matter? Maybe not. Mathematical creatures are whatever they are, no matter what we call them.

(A rose by any other name would have the same fractal dimension.)

Still, it’s hard not to want our names to reflect the symmetry and structure of the mathematical world.


Now, when it comes to imaginary numbers, you may say I’m a dreamer.

But I’m not the only one.

I hope someday, you’ll join us.

And the world will know i is “real” as 1.

27 thoughts on “Imagine All the Numbers…

  1. Imaginary numbers are the brick wall I came up against at school; I just didn’t know how to conceive of them. If they weren’t on a number line, where the hell were they? How were they used? At least with negative numbers I could say that the number/value of the things they represented was decreasing. But negative roots went against everything I’d been taught up to that point and I was stumped. (For what it’s worth, I found my scores in maths improved after I started studying physics – I guess I needed some sort of tangible, real-world application to help me understand it all.)
    A couple of decades later, and it still bugs me that I don’t know how to visualise imaginary numbers, or how they’re used. How do you explain them to your classes? Are there other learners who find them hard to grasp?

    1. You can treat imaginary numbers as the y-coordinate on a plane and the real numbers as x-coordinates — so that the (real) number line becomes a (complex) plane. That’s the origin of the “orthonumber” terminology. I like to say that i is at right angles to reality. 🙂

      1. Thanks Tim – I think this is getting me closest to expanding my mind a bit! The navigation analogy is the best, clearest explanation I’ve read.

        It’s also helping me understand the logic behind sine/cosine/tangents (which have until now always been nothing more than a button to press on the calcuator at the right time).

        Great page – bookmarked! 🙂

    2. You can visualize the imaginary numbers as being laid out on a line similar to the real number line, except at right angles to it. The relation between the positive real number x and the corresponding imaginary number xi is that x is a certain distance from the origin along the real line, and xi is the same distance but rotated 90 degrees. So multiplying by i rotates a positive real number by 90 degrees, and multiplying by i again rotates it another 90 degrees onto the negative real line, which is why i^2 = -1.

      Complex numbers are arbitrary points in the plane defined by the two number lines.

    1. Tetragons are quadrilaterals, but we really should just bygons be bygons. 🙂 Sorry for the pun. And the math object would likely be “bigon”.

  2. Imagine i and mu, I do,
    I think about them day and night
    (Why have a fight?)
    “Real” and “imagin’ry”
    Are they just text?
    The real and the complex?

    (Hey, it’s 8 AM and I’ve not yet finished my coffee)

  3. I believe that Euler called them “lateral numbers” due to their lateral distance from the real number line. I’ve always disliked the term imaginary myself.

  4. I can say with certainty that the misnaming of complex numbers with 0 real part (i.e. imaginary numbers) is still causing much confusion among educated adults even today. I had this very discussion about a month ago! It can be hard to explain that all numbers are “imaginary” in the colloquial meaning of the term, rather than the technical.

    I tried to show that you could construct the complex number from the reals by using tuples and appropriately defined addition and multiplication. Given that, how can you say they aren’t real? The audience wasn’t used to that approach and it didn’t get through.

  5. I stumbled upon this a week or two ago:

    “Imaginary numbers are real”

    It starts with a very basic introduction and works its way up to Riemann surfaces, and is worth checking out.

    The creator of that piece argues for “Lateral Numbers” rather than “imaginary.” I think we all agree that imaginary is a horrible name, yet it seems like one we are stuck with. Complex is better, but still leaves them feeling more mysterious than they need to be.

  6. When I was quite young (2 or 3 years old) I had an imaginary bunny. Not a stuffed bunny or a velveteen bunny. Imaginary.

    Then my older brother stole it.

    Again I remind you that the bunny only existed in my mind, and had no physical component that he could take. But I was still upset enough that my parents had to step in and force him to give it back.

    Years pass, eventually I get to imaginary numbers in school. When I mentioned at home that I didn’t get imaginary numbers because they have no real world component my parents reminded me of this story. They pointed out that even though the bunny was imaginary it was real enough to me that it followed the same kind of rules a real bunny would.

    Fifteen years and 2.5 collage degrees in mathematics later, that explanation of how complex numbers fit into the number system is still the best one that I have heard. It may not be the most complete one, but its intuitive nature makes it the best.

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