A Visual, Intuitive Guide to Imaginary Numbers

In your video, there were the 3 follow up questions in the end, do you have the solutions to those? Thanks and love your articles :slight_smile:

Btw, ā€œOr anything with a cyclic, circular relationship ā€” have anything in mind?ā€ Didnā€™t quite get thatā€¦

Hi,

Why donā€™t we just call these numbers by x and y instead of real and imaginary. For example say 3x+4y instead of 3+4i.

If the concept of the number line embraces zero, negatives and irrationals, and indeed even makes the relationship between them obvious, then why not extend the number line to the number plane to include imaginaries.

I am troubled by the invocation of rotation as being essential to understanding complex numbersā€¦it seems like pulling something out of your hat. Algebraic operations along the number line simply move back and forth along the line. Why not consider operations in the complex plane simply operations that move about the plane, with rotation simply being one manner of moving about the plane

Hi Raj, great question. In hindsight it seems we should just use x and y, but the notion that imaginary numbers were 2-dimensional wasnā€™t thought up until decades after their initial discovery. The name when these strange new numbers came about (ā€œimaginaryā€) stuck, unfortunately.

Awesome! Mind blown! Thanks! I donā€™t know how much good itā€™ll do, but Iā€™m posting a link to this on my Facebook page, and may later put it on my website.

Really coolā€¦i donā€™t understand how my teachers couldnā€™t get this through like this. Thanks so much for the site. Itā€™s really easy to understand.

Why is the number one multiplied in the 1*X^2=9 problem? It seems uber-redundant and something that is meant to confuse people needlessly. To me, this whole imaginary number thing seems much like religious doctrine, which uses peer pressure to get people believing in something that was made up. Even after going through many different websites trying to find different and more thorough explanations of imaginary numbers, I am still seeing it as bullshit and the more you try to explain it, the more it seems like tooth fairy B.S. or like Jesus turning a couple fish into food for thousands. It also seems that even the guy in the video doesnā€™t really understand it, as there are far too many dead-ends in the explanation. My running theory is that everything is just made up and forced down peopleā€™s throats until they just accept it as truth. Saying (ā€œjust pretend i existsā€) is not far from pretending that the tooth fairy exists and me, as an adult, put a tooth under my pillow and wake up to find my tooth still there. I donā€™t see the ā€œiā€, note do I see the need to include the number one in the equation x^2 =9. I am not trying to bash your explanation-I just really want to understand. Nobody has delivered a satisfactory explanation thus far. Thanks for trying though.

Can you do a follow up article about quaternions?

Hi kalid.
I figured out something and wanted to share with you. I used to always wonder why complex numbers come into electronics. Am an Electronics and Telecommunication student.
I figured out that complex numbers come into maths whenever one force splits into two. Rotation is just one thing that makes one force splits into two. One becomes the real part and the otherā€¦the imaginary part. A capacitor for example does thisā€¦it exerts two forces on the electronsā€¦it delays them by storing them like a dam and the leaking it after a certain thresholdā€¦plus it exerts a force of its own internal resistance.
The resistance of the capacitor is the real part of the capacitor and its storing abilty is its imaginary part. It took time for me to understand it since there was nothing rotating in this whole process.

Have written in detail in the following article.
http://visualizingmathsandphysics.blogspot.in/2014/08/why-do-complex-numbers-come-in.html

Binnoy
http://visualizingmathsandphysics.blogspot.in/

So Iā€™ve read comments down to number 100ā€¦ Iā€™m not sure if the explanation would have worked for me. I prefer this:

  1. There is no square root of -1 in the real numbers.

  2. You can define laws of ā€œadditionā€ and ā€œmultiplicationā€ to operate on ordered pairs or tuples (a, b) of numbers in such a way that tuples with b=0 behave exactly as the corresponding real numbers do, and (0, 1) x (0, 1) = (-1, 0). i.e. complex numbers.

  3. Of course you can plot these tuples as points in two-dimensional Cartesian geometry. And of course this looks just like your diagrams.

  4. Polar coordinates.

  5. Everything else about complex numbers.

This possibly isnā€™t so good for students whose mathematical intuition is geometrical. I like numbers. I like that Cartesian geometry does all the things that Euclidean geometry does because tuple coordinates and number sets -are- ā€œpointsā€ and ā€œstraight linesā€ which fulfil the axioms of Euclid and therefore fulfil all their consequences, theorems and all that, although it may be unfortunate that geometry then is just a matter of algebra which you can still do without a special geometric insight. But, back to the positive, it disposes of anxiety - mine anyway - about just insisting that an equation has a solution when it doesnā€™t (in real numbers), and the apparently meaningless difference between i and -i, which looks like it should be hugely important. For me, good mathematics is logical, and you canā€™t have things just because you want them. So, my way, you just observe the properties of (0, 1), and call it ā€œiā€.

For quite a while, it felt to me as though someone could discover an argument that completely destroyed complex number mathematics. Iā€™m much happier starting basing it on Cartesian geometry and good logical rules for operations.

***I meant ā€œnorā€ do I see, not ā€œnoteā€ do I seeā€¦in the third to last line of my comment.

Hi Alex, the reason for writing x^2 = 9 as ā€œ1 * x * x = 9ā€ is that it makes it clear that each ā€œxā€ is a transformation on the number 1.

In a similar way, we might write ā€œ-3ā€ as ā€œ0 - 3ā€ to realize that we are ā€œthree below zeroā€. Once you are familiar with imaginary numbers you donā€™t need to do this, but the first time itā€™s helpful to untangle what is happening.

Thanks for your answer. I get how we might write ā€œ-3ā€ as ā€œ0-3ā€ to realize a position relative to zero, but I donā€™t get where the number one comes from in "1 * x * x = 9 ". I donā€™t know what a ā€œtransformation on the number 1ā€ is. Am I missing some rule that didnā€™t get explained to me earlier on? I think itā€™s possible that I am lacking an awareness of some such rule(s), as math teachers Iā€™ve had either did not understand the maths themselves or purposely left out very important bits of info. I vividly remember going through college level math courses (just 3-4 years ago) and the math those teachers taught is completely different from what Iā€™m learning on your site and the Khan Academy.

No problem, happy to clarify. A lot of math education just gives you facts without really sharing ways to get the ideas intuitively, which can be really frustrating.

I see any number, like 14, as a scaled-up version of 1. 14 is the same as 1, itā€™s just 14 times bigger. -5 is the same as 1, except itā€™s pointing the other way (negative) and is 5 times bigger.

This article explains more about seeing numbers as ā€œtransformationsā€ on the number 1:

http://betterexplained.com/articles/rethinking-arithmetic-a-visual-guide/

When we see an equation like

x^2 = -1

it looks really strange until we walk through what it means. Implicitly, every multiplication means we start with the number 1 and do something to do it.

So, x^2 is really ā€œ1 * x^2ā€ just like ā€œ3ā€ is really ā€œ0 + 3ā€. We want to remember that we had a starting point.

x^2 is really just x * x, so another way to put it:

ā€œx^2 = -1ā€ is asking the question ā€œIf we start at 1, can we make two transformations, and arrive at -1?ā€

I didnā€™t say ā€œmake two multiplicationsā€ because that wording makes us think of the changes we already know. The real question is whether thereā€™s any transformation (type of change) that can turn 1 to -1 in two steps. A rotation is one such change that would let this happen. We arenā€™t used to rotating numbers, sure, just like a bug walking on a wire isnā€™t used to moving in a different direction. It doesnā€™t mean we canā€™t use that dimension though :).

Check out the visual arithmetic article above to see if it helps things.

Thanks. It makes much more sense now.

You are my hero! As many others, I used imaginary numbers through college and after reading your blog I realized that I never really understood them until today, your site is freaking awesome!

Regards
Adolfo

Thanks, this was very interesting! So basically numbers now have angles, multiplying numbers is an operation involving the addition of angles, and negative numbers have square roots. Thanks for the ā€˜ah-haā€™ moment!

My favorite equation using (i) is i^i = e^((-Pi)/2)), in other words that i^i is in the set of all real numbers. Use Eulerā€™s formula to prove it to yourselves.

From your earlier post:

ā€œThe real question is whether thereā€™s any transformation (type of change) that can turn 1 to -1 in two steps.ā€

How about two translations of -1 along a number line?
Couldnā€™t you also transform 1 to 9 by two translations of adding 5?

Why would a high school aged student see a rotation as an intuitive transformation to go from 1 to -1? You did not say ā€œmake two multiplicationsā€, but that assumption is necessary for students to follow along your train of thought.

Dear Kalid, first of all congratulations on all your amazing work, iā€™m glad someone else thinks that the approach we use to teach maths is bad, anyway i wanted to ask you something because i canā€™t find an answer anywhere on google: basically we define i as the square root of a negative number because the rules of math donā€™t allow it, hence immaginary, but my question is this: is there an ulterior motive for defining it as a negative square root? like for example could we define i as ln(-e)? being that negative logarithms are also ā€œagainst the rulesā€. I think finding that the answer might provide additional insight on immaginary numbers. What do you think?

Sincerely, another math lover