A Visual, Intuitive Guide to Imaginary Numbers

@Ajax: You’re welcome!

@Wren: I appreciate it, thanks!

Hi Kalid, tried to post a comment without any luck. Hopefully this works.

Thanks very much for the guides, I particularly enjoyed this as well as the guide on exponentials. e has always given me such a headache! Using what you taught us there (namely that e is the limit of rate of growth…did I get that right?), what can we say about Euler’s Identity ($$e^(i*pi)=-1$$)? Is there an intuitive physical meaning?

Again, thanks so much, you’re very good at what you do!

JD

Good question raised by Vernon, It is well know Einstein’s passion of Maxwell’s equations in electromagnetic field and energy. Add to that his obsession of speed of light since he was a kid.
I think it is the energy (or more specific, kinetic energy) that led him to link mass to speed of light.
This video http://www.youtube.com/watch?v=hW7DW9NIO9M explains why (m) suddenly appeared in Einstein’s equation. Or you may hear it from Einstein himself http://www.youtube.com/watch?v=vb1EO6aoaFQ

The way I like to think about it is that “i” allow us keep the reverse of an exponent just like negative numbers allow us to keep the reverse of an addition. For instance,
let’s say x=sqrt(9). I can express x as being equal to sqrt(-3*-3). Then I should have the right to write x=sqrt(-3)*sqrt(-3) just as I can write x = sqrt(3)*sqrt(3) no?. But without imaginary numbers, this doesn’t work. I can’t “store” a negative square root, just as we didn’t have a way to “store” a negative result before negative numbers came in existence. So in order to keep algebra consistent under the exponent function, we had to come up with the imaginary numbers. Otherwise, we would have ended up with inconsistent results, as in this exemple, depending of the actual operations used to solve the problem.

Thanks Alexandre, that’s a great way to see it also.

thx disha.karnataki (so obvious!! how I missed it! getting old I guess!)

This is just a mad random thought.

Could we not use the idea of the square root of minus nine to describe, more completely, the idea of the number 3 looking at itself in a mirror.

When I see myself in a photo, I realise that my hair is parted on the opposite side when I look at myself in a mirror.

In a mirror I appear identical, except that I am a rotated mirror image.

Martin Escher’s drawing of a “hand drawing itself” may also be relevant.

Also, what happens when I hold a mirror in front of a mirror? The recursive image disappears to infinity.

As an only child, at about 2-3 years of age, I remember thinking that I had found my sibling in the mirror. Why wasn’t he behind the mirror when I looked? Was that my “square root of a negative number” moment?

I just love your website. It reminds me of how little I currently actually know and conversely how much there is still to learn and understand.

Hi. Thank you for your explanation. I actually found your site after searching for information on Pareto’s 80/20 idea, but I just could’nt help browsing your explanations of things, and boy am I glad I did. This notion of a complex number has given me nightmares. I love your take on the subject. I especially love what you said about Euler and him not even understanding negatative numbers. Did you know that when he wrote his papers he did it in a style so that he ‘did’nt have to argue with those of lesser understanding’? From what I’ve read, those are actually his words…That guy has given me too many headaches for too long. Thanks again.
By the way, and this is a bit unrelated to the topic, but I was wondering if you had any idea how that other genius of the age, Einstein, figured out how E=M.(the freakin speed of light)^2…How did he know E was proportional to lightspeed^2? When I try to relate the formula to something I know a bit about, I think of nuclear material breakdown, and it works, but how on earth did he know the formula depended on c^2! Apart from complex numbers, this is the last thing I need to know before I expire, so your assistance would be much appreciated.

[…] but I don’t want to get off track. For the mathematically inclined, I would highly recommend this excellent explanation of what complex numbers are and where they come […]

[…] but I don’t want to get off track. For the mathematically inclined, I would highly recommend this excellent explanation of what complex numbers are and where they come […]

This article can do without the incessant exclamation marks, childish outcries, and baseless foundations such as the Romans not understanding division.

@Brian: Thanks, glad it’s piquing your curiosity!

@Vernon: Awesome, glad you’re enjoying it. Many times, topics are presented in an overly formal or complex way – to appear impressive? In a misguided attempt to “wow” the student? I’m not sure.

Einstein’s original paper is only 3 pages: http://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf

And especially this statement: “If a body gives off the energy L in the form of radiation, its mass diminishes by L/c^2”

(Note, he used L and not E to represent energy originally). He basically realized that as energy was transferred, the mass changed by m = E/c^2. After some re-arranging, you get the famous E = mc^2. There’s more to it, but his original formulation was a different form of what we recognize today!

@ACG: One my favorite parts about the internet: to each their own!

[…] Negative numbers were also controversial at first.  How can one have ”negative two apples” or a negative quantity of anything?  However, it became clear that negative numbers were indeed useful conceptually.  If I have zero apples and borrow two apples from a neighbor, according to my mental accounting book, I do indeed have “negative two apples,” because I owe two apples to my neighbor.  It is an accounting fiction, but it is a useful and valuable fiction.  Negative numbers were invented in ancient China and India, but were rejected by Western mathematicians and were not widely accepted in the West until the eighteenth century. […]

[…] but I don’t want to get off track. For the mathematically inclined, I would highly recommend this excellent explanation of what complex numbers are and where they come […]

I love this and I will be sharing it with my high school students even if it makes some of them glaze over and hurt their brains. It is an excellent representation of i. thank you for creating this.

May I suggest you edit your discussion of negative numbers with the following observation: According to the http://en.wikipedia.org/wiki/ article on Double-entry_bookkeeping_system, it was fully in place (by the inventor) in 1300. Today, the confusing (to me, anyway) system of ‘Debit’ accounts and ‘Credit’ accounts seems to be an obvious holdover of a primitive and awkward understanding mathematics (at least from a modern point of view).

Imagine for a moment how much easier it would be to learn accounting if accountants learned (or accepted) ‘modern’ (post 1759) mathematics!

I really used to hate complex numbers. Now I’ve started loving them. Infact complex numbers aren’t really complex at all.

Great job man… :slight_smile:

Wow, great article. This might look incredible, but it turns out that I actually learned of imaginary and complex numbers for the first time by actually asking a question about rotations. I was asking my mom what does it meant to have a negative velocity, and she told me that it was traveling the same speed but in the opposite direction. I wanted to throw a challenge at her and at the same time satisfy my curiosity, so I said: " okay, so traveling at -5km/h North is the same as traveling at 5 km/h South. But how about the relationship between North and East? If I’m traveling at 10 m/s East, at what velocity I’m traveling towards the North?". She stared at me for a while, but then she said: "Solve this equation, and if you make it I’ll tell you, and she wrote in my notebook: $$x^2+1=0$$. Immediately I told her that there is no number such that when squared you get a negative number, and she said “Oh yes, there is, and that number is ‘i’.” I was like “what??! What are you talking about?”. Then she introduced to me the new numerical system of the complex numbers (which is not really that new). I kinda felt disappointed because I really thought she wasn’t going to be able to answer that one, and that was what I really wanted, but then at the same time it felt good I knew the answer. The next day I went like crazy to my science teacher and I asked her the same question, but this time I was testing her rather than just looking for an answer. She told me that there is no such a thing but I told her: “You’re not telling me the truth, Miss. I know you know the answer.” Then I explained to her and she started laughing and said “Oh okay you got me here.” Nowadays I spend looking for answers to all sort of different questions about complex numbers like “What’s the sine of i?” “How do I take the logarithm of a complex number?” “What meaning does ‘i factorial’ have?” or “Is i=-i?”. I already found the answer of some of them, others, not quite yet. I know the sine of i equals $$(e^2-e)i/2e$$ and that in order to take the principal logarithm of a complex number you have to use polar form. However, I still don’t know what is i factorial or don’t know whether -i=i, because -i and i cause some ambiguities. What is so funny about this is that I started dealing with complex numbers when I was 10. I’m 14 and I’m still looking for many answers. In fact, looking for these answers led me to this blog.

[…] notion of zero is biased by our expectations. Is “0 + i”, a purely imaginary number, the same as […]

@Ketchup: Glad you enjoyed it :). The answers to the questions are in this article: betterexplained.com/articles/intuitive-understanding-of-eulers-formula/

Oh, the cyclic, circular relationship was a hint at sine waves, more here: http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/.