thank you. thank you. thank you.
Just taught this to a bunch of year 14 years olds, and it helped my understanding to read it to! Thanks!
Understanding Imaginary Numbersā¦
I thought I understood imaginary numbers fairly well before today, but reading through this brilliant explanation crystalized it for me. Imaginary numbers always confused me. Like understanding e, most explanations fell into one of two categories: Itāsā¦
[ā¦] imaginary numbers as rotations was one my favorite aha [ā¦]
[ā¦] Why Complex Multiplication Works | BetterExplained. Seeing imaginary numbers as rotations was one my favorite aha [ā¦]
[ā¦] Articles | All Posts A Visual, Intuitive Guide to Imaginary NumbersA Visual Guide to Version ControlMental Math ShortcutsHow To Measure Any Distance With The [ā¦]
Negative numbers allow you to flip in one dimension. Imaginary numbers allow you to enter into two dimensions. What comes after that for 3, 4, etc. dimensions? In a sense we can do any number of dimensions using x,y,z,etc. coordinates, but thatās not exactly what weāre talking about in this article.
If there isnāt anything like that then why is that? If the answer is that matrices and/or x,y,z coordinates are good enough for additional dimensions then why is i necessary for rotations as described in this article?
@Geoff: As you say, negatives are 1-dimensional ā and only invented in the 1700s! They opened our mind to whatās possible. Imaginary numbers can later (2d), and I think people realized you could have any number of dimensions in linear algebra / matrixes (and didnāt have to visualize them).
For reasons I donāt understand, I donāt think you can have 3 dimensions easily (due to symmetry) but there are 4-dimensional numbers (1, i, j, k) called Quaternions (http://en.wikipedia.org/wiki/Quaternion) that are actually used in video game programming to model rotations! There are even 8-dimensional numbers tooā¦ though at some point, it becomes easier to just use a matrix.
I think the neat thing about āi are rotationsā is that it expanded our mathematical perspective to āacceptā that numbers can be 2d. It would be hard to really believe that numbers could have two perpendicular components without an analogy like that. After i broke the ice, we can accept that numbers can have any number of perpendicular components :).
Thatās my 2c on it anyway!
oh, shit, how come I did not know this 20 years ago? It was very very frustrating, which even made me frustrated with math from high school to uni, even today, but now i am released.
unlimited thanks to your explanation for ever. If given any chance, I will share your comments with as many others as possible.
@johnway: I had the same reaction ā only understanding them way after ālearningā them.
[ā¦] as well for you to take a look at. Ā Here are a few links I have collected over the last few weeks.Explanation of imaginary numbers. First of all I found an excellent explanation of whatĀ imaginaryĀ numbers are. The website is [ā¦]
absolutely fantastic explanation. I am a college lecturer and always thought that i was the absolute best at explaining things( I teach finance and accounting and taxation). I am impressed to say the least. I was pretty good at maths in school. but dint really understand a lot of things. this is the first time i have understood the concept of complex numbers. I used your explanation to explain complex numbers to my 13yr old daughter and she also thought that the explanation was awesome. Indebted to you for life. thanks
@sharad: Wow, thank you for the wonderful note! Iām happy you were able to share the learning :).
Problemen:How can I find: (i + 1)^i
Thank you so much for that. I actually ENJOYED reading that, and math does not usually ātoot my hornā.
@Thorsten: Awesome! Thatās the most I can ask :).
Thanks a lot for the great article simplifying the abstract idea.
About the last note you wrote, Iām also fascinated by the invention of number 0 and I think it might be a cultural concept in India before it became mathematics. The ancient india believe in the Ether element that is void but upholds the physicality of the other elements. The representation of that is probably 0 , and when big numbers are represent as 100 the zeros are not actually there but only upholds the realness of the concept of hundred.
Wow!!! This was a very interesting and amazing explanation of imaginary numbers. Iām 38 years old and always love to study science, technology and math. However, itās been a while since I donāt get into the topic of imaginary numbers and this helps me a lot to ābreak the iceā. Congratulations!