Homework: I understand that (i) is similar to 0 and that it’s a weird way of making math easier. I understand WHY it was created but I’m having a littel difficulties with understanding exactly what it represents. I know that the sqaure root of negative 64 is 8i but I’m confused as to what the (i) stands for. This helped me understand though !
@Cheryl: Sorry to hear that – please feel free to leave any questions. The main insight is that numbers can be 2-dimensional. In most circumstances, 1 dimensional numbers are fine (money in my bank account is positive or negative), but sometimes we want to consider things in multiple dimensions (such as the trajectory of a ship).
@Mr r: Thanks!
Nice one.
Hi Kalid, I wanted to let you know that, after further exploration, I am at now at peace with my level of understanding the imaginary number. Thanks again for the insight.
However, when you have an opportunity, I am still interested in your thoughts on number 2 in my “Alonzo on March 13, 2012 at 1:11 pm” posting about multiplication vs addition.
Thanks,
Alonzo
Brilliant explanation!!! This is by far the best and most helpfull math website, I have ever come across.
However, I have a doubt. Earlier, in your rotation example, you rotated 3+4i, 45 degrees by multiplying with (1+i). while I understood this, I am thinking, “why dint he multiply by i/2”? if " i " rotates a heading by 90 degrees, the surely, “i/2” should rotate it 45 degress.
i did calculate (3+4i) . i/2. Turns out, the new heading is 90 degree rotated. but the magnitude was halved. Still not getting why this is so.
I still don’t understand 
@123: You’re welcome! Good luck with that report.
@Jaycee: Awesome, so glad it helped! Getting imaginary numbers to click was one of my favorite moments in all of math.
Hello Kalid, let me first start by thanking you for taking the time to provide a detailed response to my questions. Your insight not only helps me, but as a consequence it helps me when I try to explain things to younger folks in an attempt to keep keeps their interest and confidence with math (not a teacher nor a Math person, just someone who has seen one too many kids completely give up on math). Here are my follow-ups to your response.
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You wrote, “Similarly, 1 and 1i are both “1 unit” away from 0, but in separate directions.” So is this to say that i is the unit of measurement on the imaginary axis ? And with magnitude being a measurement of distance with respect to an origin, we get - in this case of ‘1i’ - the measurement of a 90 degrees rotation around an origin ? But, then is that to say that the units in the imaginary and the units on the real axises are equivalent - thus the units on the real axis are also based on 'i’s unit notion of rotation? If this is so, then I think there may be hope for me. If not, then wouldn’t we have 2 axises with different units in a way similar to say… feet on one axis and miles on the other? In these types of scenarios, don’t we have to convert one of the two units of measurement to match the other before we can do things such as calculate the magnitude of a vector/hypotenuse using the Pythagorean theorem (or at least account for the difference in units somewhere in the calculation)? So, if the units on the real and imaginary are (or can) be different, then how/when does this conversion occur with the use of complex numbers ? Perhaps the key to my confusion lies in a misunderstanding of what a “unit” is (but I am pretty sure I have the basic idea) and the rules invoked when calculating numbers that have different units. I can understand the scenario when you do something like (multiply 2 dogs by 7 cats = 14 dogs and cats), but if I had 20 miles on an x-axis and 20 feet on a y-axis, it seems like I would have to convert one of the two units, so that they are equivalent, at some point in the calculation of the magnitude.
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Thanks for giving me a non-symbolic definition for multiplication. Your answer is exactly the type of response I was interested in. I think one of the interesting aspects of your description is the use of the term “how” in the following, “how to apply the “essence” of one number to another?”. This question of “how” almost assumes that the essence of the two numbers can take on an interaction between each other. Do you believe your description can extend to concepts beyond numbers ? Speaking of extension, to extend this conversation even further, how would you then make a similar abstraction for Addition? What would you say are the distinguishing characteristics between the two abstractions such that they deserve their own term ? With the assumption that your abstraction of multiplication can be extended beyond numbers, and without yet having your abstraction of Addition, how would you categorize the interaction of 2 chemicals ? Would you say the essence of a mixture more so mimics Addition, Multiplication, or perhaps one followed by the other ? Assuming I interpreted you description of Multiplication properly, perhaps an example of Multiplication vs Addition could be found in nature: When reproduction occurs, Multiplication takes places (genetically) and the result is an Addition to the family. So in this sense, the interaction from the properties of the parent’s genes had a direct affect on the resulting offspring (Multiplication), and the Addition to the family (although grew the family) did not affect (at least genetically) any previous “Additions” that the family already had (i.e older sibling) … Well, maybe you have a better analogy cause it doesn’t work so well against the concept of Negative.
Thanks again for the response !
@Arien: Awesome, glad it was useful.
@mr c: Thanks for helping out!
Actually, the dogs and cats thing does not make sense to me. Why would you multiply 2 dogs by 7 cats. There seems to be no reason to do this (by the way, adding dogs and cats does make sense - thank goodness 
). Not sure where my head was at…perhaps a better example would be 4 rows times 5 people/row to get 20 people. But here the units cancel out and it makes sense to me. Oddly enough, the bizarreness of the cats and dogs scenario is equivalent to the bizarreness I find of the possibility that the real and imaginary units are not the same, nor are ever adjusted, for calculations such as magnitude.
Awesome! Mind blown! Thanks! I don’t know how much good it’ll do, but I’m posting a link to this on my Facebook page, and may later put it on my website.
http://www.facebook.com/moreinsanity (Feel free to LIKE the page)
http://www.mopjockey.com (Feel free to FOLLOW the site)
Really, man, I haven’t felt this amazed in a long time!
Hello Kalid, thanks for taking the time to share your explanations of mathematics. I have two questions based on the teachings and comments above.
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This one his hard for me to articulate but here is my attempt. Under the section “Understanding Complex Numbers” you wrote, “We’re at a 45 degree angle, with equal parts in the real and imaginary (1 + i).” But, there is no clear explanation as to why (1+i) necessarily equals 45 degrees. I could see this if the magnitude of i equals the magnitude of 1 (because you could show a 90 degree triangle with two sides equal to 1, that is an isosceles right triangle). But, saying that the magnitude of i equals the magnitude of 1 seems inconsistent with the rest of the teachings. So what am I missing ? Maybe I am actually confused with i altogether - is it meant to represent the concept of rotation around an origin (similar to a negative sign representing the flip across a vertical axis) or is it a type of number (imaginary) which is said to be the square root of i^2, or is it meant to represent both with the use of only one symbol ?
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In one of your comments above you wrote, “Multiplications can do two things: scale (change the size) or rotate (change the orientation).” Given two different descriptions of the term “Multiplication”, I assume it is intended to say that these are two interpretations of the term Multiplication. So then, what do the two interpretations have in common that maps back to an abstract definition of Multiplication from which all interpretations adhere to ? In other words, one asks - “So then, what does Multiplication in its purest form mean ?”. Maybe it is me, but with something as exact as mathematics, I can not stand the idea of having one term with two different interpretations, yet nothing concrete that drives the concept behind both of them.
Thanks
This was awesome. You are great teacher.
great article Kalid - for the first time I feel I understand the damned things (which probably means I don’t 
- though I’d heard the rotation aspect before I hadn’t really understood it. Your style of explanation is excellent - I like the visual aspects of your methods, that helps a lot with me, and your conversational style keeps it light whilst not drifting off-topic.
One thing that puzzled me was why adding 45* - whilst it was only talking about direction and not quantity - increased the quantity (the distance form zero) from 5 to just over 7 - I couldn’t get my head round why that should be, and a word or two of explanation might be helpful there (unless you consider it out of scope).
“Today you’d call someone obscene names if they didn’t “get” negatives.” I suspect you will laugh yourself silly (or cry) at something that happened here in the UK a few years back: see http://menmedia.co.uk/manchestereveningnews/news/s/1022757_cool_cash_card_confusion !!!
Thanks again for this site, a great resource and I completely agree that “it frustrates me that you’re reading this on the blog of a wild-eyed lunatic, and not in a classroom” - if only we’d had a lunatic like you teaching our maths class!
[…] you get that imaginary numbers are numbers in another dimension, it’s about 10 minutes until you have genuine interest in […]
Thank you for this nice article and for making your ideas public with this website. Everything I always thought about learning math, but rarely found, I found here.
I always thought that the kind of understand I wanted to have was a ideal thing, that only a few people would look for.
And as I study medicine, I dont have time and energy to go through the process of learning on the formal way, and them get to “that” undestand I was looking for.
Thats why its good to read your articles, you do the hard way for us , and bring it the way its easier to understand, something that all teachers shoud try to do.
Its very good to know that there are more people, and very intelligent ones, and also graduated, that think like me.
Really, congratulations for this site. And continue this work you are doing, because as it did to me, i am sure it helps the math learning, and contribute for creating more enthusiasm for knowledge for young and even older people who was educated to see math as something cold and static.
Ok, here is my question. If the angle wasn’t 45º, but was like 58º, you would have to use pythagoras to discover the x and y of X + Yi, wouldn’t you? It wouldn’t be 1 + i
Then even with imaginary, we still depends on pythagoras, am I right?
sorry about my english, i am from Brazil
I must be Thick! or have petrified thinking ( I’m 65) . If i can be any angle .
How can i +1 be an angle of 45 degrees. Please don’t groan!! I know I’m missing the concept especially since Ethan notes that its obvious…