I love your site. Have you ever noticed how the math rules change for j? +j * +j = -, -j * -j = - and only +j * -j = +. Also how the pattern is the same as charge rules: + and + Repel, - and - Repel but only + and - Attract? I have an idea about the two thoughts. If you want to hear it, please let me know. Thankyou for your time, Angela
Hi Ninji, check out http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/. In short, when taking a complex number to a complex power, you “rotate the rotation” and end up back on the real number line.
what is Complex number to a complex power ??
I thought I understand the complex number after seeming your explanation.
However, my thought stuck again after I see i^i …
What does this really mean in turn of rotation and magnitude
I refer to the comments from various people on notation i.e. aib instead of a+ib. When it comes to a negative number on the imaginary axis you would need ai-b, so it is not workable. You could not put the sign forwards as in -aib as this gives a negative on the real axis. The notation while not intuitive for math symbolism, is sufficient for applications. a+ib points up and a-ib points down, a negative a points back and a positive a points forward. I like the notation as it shows me where a phasor is pointing. I study engineering and we use j in place of i, but it’s the same stuff.
Hi Kalid,
your explanation and your blogs are really impressive, for me it is like i found gold treasure, as i too like to understand maths and science same way you do…
Regarding the complex number, i have few questions, it will be grateful if you could clarify my points.
- I could see sine wave forms are expressed sometimes in real time and sometimes in complex way/format. what are the situations/places they use it differently. when the conversion happens from real time sine to complex sine, what is the change? how to understand it intuitively? whether conversion is possible or not?
- in Laplace transformation, e^(-st), why is -st only? why can’t it be +st?. is there any intuitive reason for this?
may be i am asking too many questions, but these questions were haunting me for long time but couldn’t understand intuitively…
Thanking you,
Upendra
I am Maths teacher for over five decades. My students ask me “why complex numbers”. I tell them if we stop our study of Maths with real numbers then we cannot solve the quadratic equation ax^2+bx+c=0, when the discriminant
b^2-4ac <0. This is the most basic necessity. l
Hi Ian, glad to hear you’re enjoying the site!
The wording probably wasn’t clear enough, thanks for the feedback. I meant that if we start with a number x, and multiply by z, we need to multiply by z* and then divide by |z| * |z| to get our original back.
For example, if x = 2 and z = 3 + 4i, then
x times z = 2(3 + 4i) = 6 + 8i
x times z times z* = 2(3 + 4i)(3-4i) = 50
x times z times z* divided by |z| * |z| = 2(3 + 4i)(3-4i) / (5 * 5) = 50 / 25 = 2
Hope that helps!
Hi Isaac, thanks so much for the note. I started the site to help other students and it’s really gratifying to hear when it’s helping. Finding the genuine intuition behind a concept is inherently enjoyable and makes the idea seem natural – we’re experience the concept directly. I hope to share this viewpoint with others over time. Welcome to the site!
Hello Kalid,
I have read at least 10 of your pages so far unraveling the mysteries of math. It turns out these ideas aren’t so mysterious when expressed the way you express them. I truly value your contributions by creating these pages, and genuinely express thanks for taking the time to share your thoughts and ideas in a clear and easy to understand way.
I am currently a first year engineering student, taking all the maths basics: calculus, linear algebra… with more maths study to look forward to in the coming years. I have no problem memorizing the formulas and understanding the ideas taught with memory tricks, analogies, and of course practicing lots of exercises from the textbook, so that the ideas stick. It’s a decent textbook too written with a somewhat intuitive feel. But I feel you have uncovered something deeper, which is more like a completely natural and sensible approach to mathematics. When interpreted and explained naturally, mathematics makes so much more sense. Before reading your articles, I had the idea pasted in my head that integration was simply finding the area bounded by some defined curves, or finding anti-derivatives, or whatever. But your explanation of it being more like a “better multiplication” of one quantity by another quantity really struck me, and integration described in that way really makes it click.
I enjoyed this article too, and the others I read. The idea that numbers are 2d and i represents a rotation. It is sad that I had to stumble upon this website after randomly google searching who knows what (I can’t remember). They should really be teaching mathematics like this in schools from the very get go. Looking at mathematics through your eyes it begins to look more like a beautiful way of describing nature itself, rather than the rigorous formulas and rules and definitions and proofs. The “why” of it all is very quickly becoming crystal clear. More students would definitely be able to catch on to mathematics and actually appreciate it if they learned these intuitive ideas, instead of some of the material they make us absorb and regurgitate come exam time. Anyways, I’m sure your articles will make a hugely positive impact on my studies, and seeing mathematics in a new and intuitively guided light will definitely make for less headaches and more “aha” moments on the road ahead. Coming across this website just might be the greatest contributor to my overall maths studies. It really makes maths seem like an upward elevator ride with something new at each floor, rather than some complicated and difficult mountain to climb. This approach should definitely be more universally taught.
Hi Kalid!
I love your site and your approach to understanding math concept. It has opened my mind. You definitely figures on my ‘inspirational people’ list.
Just one thing that bothers me here: [A Quick Example] (…) "One caveat: with conjugates, you need to divide by |z| * |z| to remove the scaling effects as well."
Re(zz*)=|z||z| and Im(zz)=0, isn’t dividing just by |z| enough in this case as we just want to cancel scaling with |z| effect? Correction by |z| * |z| is understandable for me while dividing two complex numbers:
scaling (*|z|) ->neutral (*1) ->shrinking (*1/|z|)
Keep up the good work (:
thanks, this was really helpful and I now understand.
Kalid, I like the series you have posted. It is important to provide the intuitive understanding of such stuff.
I also intend to explain these concepts to others in the same way.