Hi Weston, happy you enjoyed it! Yep, imaginary numbers can be considered a type of vector, with certain rules for how to multiply & add them.
You can model imaginary numbers using vectors and matrices (more generally in the field of linear algebra). They’re actually a neat way to introduce the topic of keeping more than 1 piece of information in a single “number” (or vector 
).
-Kalid
[…] Imagine - imaginary numbers! What could this mean? Something made up by mathematicians with way too much time on their hands? What are these folks going to think of next? Algebra teachers have long known that equations of the form x2 = 16 have two solutions, +4 and -4. But when asked to solve the equation x2 = -16, most people say “there is no such animal” that can be squared and equal a negative number. However, we can now find the answer to this, and other questions, and see this answer with great understanding and visualization by visiting the website of Kalid Azad. He has created a website, BetterExplained, with the byline “Learn Right, Not Rote“. His goal is to help people understand mathematics with incredibly lucid explanations. In the case of imaginary numbers, he focuses on relationships, not mechanical formulas, presents complex numbers as an upgrade to our number system, just like zero, decimals and negatives were, and uses visual diagrams, not just text, to understand the idea. His secret weapon is learning by analogy. He approaches imaginary numbers by observing its ancestor, the negatives. Now Algebra 1 teachers can explore imaginary and complex numbers with their students with understanding and appreciation. Enjoy Kalid’s website and let me know what you think. […]
THANKS!i found your articles very very useful.
Outstanding! I never thought of the complex plane in terms of rotation before. I’m teaching myself 3d graphics, and I’m not satisfied with “put the model in the engine and don’t worry your pretty little head over how it works”, and one area that is blowing up my head is in the area of quaternions. This might help crack that area open for me.
… But on the off chance that I or others still can’t get any traction on the topic, you think you could cover quaternions in a future article?
@vaibhav: Glad you enjoyed it.
@chuck: Thanks! Yes, I’m never satisfied with “plug and chug” either. Sure, I’ll add quaternions to the topic list – no estimate on the ETA though :).
[…] The number line is two dimensional? (You bet — imaginary numbers) […]
Hey, I’ve got a few questions/comments myself.
i
ANS ^ ( 1 / i )
ENTER/SUBMIT/EQUALS (whatever it is) Three Times
You get a seemingly weird result. You want to explain that one geometrically?
–
Having no real knowledge of what ‘i’ is, never knowing more than it’s the result of the square root of negative one, I had the thought of drawing a geometric plane with one axis being the real number set and the other axis being the imaginary number set, as you showed above. I showed it to the Pre-Cal teacher (as I was in a programming class she taught) and she got confused by it. This was two years ago, in a different school, when I was High School Freshman. What took me a few hours of actual thought on it (counting time sleeping, about 15), I managed to recreate something that took most mathematicians a few years to grasp. Sadly, I was rejected when showing the idea…but you’ve turned my thoughts back around to proving me true.
–
A point on saying that numbers have two dimensions. I’m guessing you didn’t take the next step for simplicity, and just say that numbers have an infinite amount of dimensions.
hi,
thanks a lot for these explainations,
i start to understand it better,
i must say these imaginary numbers have been a full stop to my scientific studies about twenty years ago,
turned towards languages sports & phylosophy for my graduation,
really messed my studies after that, gave it all up
to professional life,
been musician,
actor,
dishwasher,
teacher,
hobo,
psynurse,
sailor,
autor,
poet,
till the day i found out all that not satisfying
as some light missing,
really loved science as kid so turned back to those teachings,
& making an aim of understanding those imaginary numbers,
intuitivly thinking they where the key i needed for that door,
the candle to light so i can take the next step
thanks a lot
^-^
lol thx helped my curiousity im only in 9th grade >.> but it now my brain doesnt ponder about wat it is.
Thanks for doing this, I totally support what you’re doing. I don’t understand why understanding is so seldom taught, if not consistently assaulted. Several cynical possibilities spring to mind.
I think so many people are cut off from the beauty of the mathematics as a result. For me it’s like having been deprived of a sense for most of my life. We would not think it fair to stop someone being able to smell or taste, nor is it fair to deprive them of those beautiful moments of silent, wordless understanding, the "a-ha!"s. Our joy consists in coming to better know a piece of nature.
I’ve always been interested in physics, but I can’t go any further without more mathematics; something I’ve always found difficult. What a chore, I thought. Then I had the revelation that maths was beautiful if one tried to undertand it like physics. It’s almost physics in reverse, unpacking the maths into intuitions, rather than packing intuitions into maths.
Wikipedia can be disheartening because it often comes across as preaching to the choir - showing off to people who already know the subject and wouldn’t need to look it up anyway. What good is the sum of all human knowledge if you can’t understand it? What you’re doing here is really valuable.
@Psy: Thanks for the comment! I agree, I don’t know why understanding (not memorization) is the focus – I suspect it’s because memorization is easier to test.
I like that point about the beauty of math and physics – many people scoff that notion, but the beauty really is there! I don’t know how to explain it either, but the way such simple rules can create something so complex and powerful is mind-boggling.
Also agree on Wikipedia – it’s a great reference, but not a good learning aid as it’s often written at the most detailed level (by experts, for experts). Ideas die unless they can be understood and learned by later generations. Thanks again for the comment.
hi kalid,
thank you very much. i have not found any better explanation than this one on complex numbers.
– abu ihsan,
Kuala Lumpur, Malaysia.
Thank you for your beautiful explanations.
OMG!!! this made so much sense, and was very very easy to follow. it wasnt very mindblowing, and it actually made sense!!! i like having answers to why i is i and all that jazzz…
thanks a ton!!!
In high school my teacher could not get me to believe that there could be any such thing as the square root of a negative number. Your site has made me a believer!
@Ti: Glad you found it helpful!
@Daniel: Thanks, I didn’t “believe” in i for a long time either :). Only when I started accepting that numbers could have more than 1 dimension (why not?) did it click.
Hi,
Thank you for light a candle
[…] Other ideas aren’t so lucky. Do we instinctively see the growth of e, or is it an abstract definition? Do we realize the rotation of i, or is it an artificial, useless idea? […]
Hi Kalid, great article - I wish I had something like this when I was learning it in school. Believe it or not, I only understood this when I was actually working on complex power as an electrical engineer. Yep that’s right, I was being paid and didn’t even realise that a complex number was just a number with two dimensions. I could do all the algebra, but didn’t understand the fundamentals. Scary thing is, most engineers I know don’t really understand this either… it’s just not taught properly.
One thing you could add is something brief on complex exponentials, which is another thing I never understood properly at uni. Euler’s law is usually just taught in passing on the way to Fourier transforms and series. But it is seldom derived and I thought it was magic for a long time! I personally found the derivation based on the Taylor expansion of e to be the most intuitive.