31–40 of 143 posts
Imaginary numbers are defined. Imaginary numbers are not rotations or anything else that I keep hearing. Those are all properties of the fact that we define an imaginary number to be z = a + bi, where a and b are real numbers, and i^2 = -1. That's it. If someone asks what is an imaginary number the correct answer is "z = a+bi, where a and b are real numbers, and i^2 = -1".
Here is an even better discussion on the same topic, and the HN thread from last year: http://betterexplained.com/articles/a-visual-intuitive-guide... https://news.ycombinator.com/item?id=2712575 One great conclusion from this approach is how intuitive it becomes to understand the square root of i . I always thought you'd need another dimension to describe that, and another dimension for the square root of that unit,…
Read about the roots of unity (solutions to a^n = 1) to see the same concept in action. http://en.wikipedia.org/wiki/Root_of_unity
It all comes out of the properties of rotation. We can first show that there are n nth roots of unity. It comes down to a^n = 1 -> a^n * a^n = 1 -> (a * a)^n = 1, which effectively shows that any multiple of the angle represented by a will also be a solution. If you use the full fledged exponent formulas, it's easy to see that there will be only n of these. Then if b^n = i (our case), b^n * a^n = i * 1 -> (b * a)^n = i. So adding any of the nth roots of unity to b will get a new solution for the nth root of i. Since one of the roots of unity is always 1, aka an angle of 0, we will only get n-1 new solutions from this. Hence there are n solutions. :)
EDIT: New here, didn't know the markup cues would interfere with my equations.
http://www.amazon.com/An-Imaginary-Tale-Story-square/dp/0691...
1. Matrices, especially matrix multiplication. Unlike matrix addition, multiplication is defined in a very weird way. I think I understand where it is coming from -- defining it that way allows representing and solving linear equations. More insights, however, would help.
2. Dot and cross products. E.g., the magnitude of dot product in 3D is a.b.cos(theta), while for cross product, it is a.b.sin(theta). I never got sure what happens to other combinations like a new "vector" product whose magnitude is a.b."cos"(theta)?
3. How is a set and "belongs-to" operator defined? Most books I have come across just assume these (and later define natural numbers and addition from them).
4. Why is 0.9999... considered to be "equal" to 1. I understand them to be equal "under the limit", but not without. This seems to be in my way of understanding Cantor's infinities.
Why does (-1)*(-1)=(+1) ? It is arbitrary, and there really is no good reason. We could construct number lines that work differently, so that imaginary numbers never appear. Such alternative number lines would still allow us to solve the exact same physics and engineering problems. Sure, the computations would work differently, but the way we would measure and use the initial conditions in our equations would be diff…
EDIT: I proved the wrong thing. Correct proof is in post below.
Here is an even better discussion on the same topic, and the HN thread from last year: http://betterexplained.com/articles/a-visual-intuitive-guide... https://news.ycombinator.com/item?id=2712575 One great conclusion from this approach is how intuitive it becomes to understand the square root of i . I always thought you'd need another dimension to describe that, and another dimension for the square root of that unit,…
On the one hand, the mathematician inside of me is raging "this is neither specific, nor rigorous!" In my opinion, this definition is as close to useless as a mathematical definition can be, since the person who has this (and only this) will be hard pressed to answer any other questions involving complex numbers. For example, what is multiplication by i+1? Without further understanding, I don't think such a question can be answered.
On the other hand, the teacher in my head is celebrating. This person managed to take a concept normally feared and hated by the population at large and made it interesting and understandable. The response is well thought out, and is much more engaging to the reader than any response that I could come up with. In short, this explanation is more likely to make the person interested in mathematics, while a more technical definition would probably just make them zone out.
What do other people here think about this? Is the clarity and interest worth the loss of utility? Am I totally off base?
Here is an even better discussion on the same topic, and the HN thread from last year: http://betterexplained.com/articles/a-visual-intuitive-guide... https://news.ycombinator.com/item?id=2712575 One great conclusion from this approach is how intuitive it becomes to understand the square root of i . I always thought you'd need another dimension to describe that, and another dimension for the square root of that unit,…
One more question; what is the "i"th root of i? Please try to use the angle metaphor :)
To get the 'i'th root we divide the exponent by i, so i^(1/i) = e^((i * pi/2)/i) which is e^(pi/2).
As for the angle metaphor, that works for multiplying by complex numbers. The usual "understanding" of exponentiation is repeated multiplication, but taking a power of "i" can't be understood as multiplying together "i" lots of the number. Similarly the "i"th root can't be understood like that.
But you knew that.
Here is an even better discussion on the same topic, and the HN thread from last year: http://betterexplained.com/articles/a-visual-intuitive-guide... https://news.ycombinator.com/item?id=2712575 One great conclusion from this approach is how intuitive it becomes to understand the square root of i . I always thought you'd need another dimension to describe that, and another dimension for the square root of that unit,…
Do kids not learn this in 8th grade any more? I'm seriously not trying to be snarky, I just can't think of a way to write that question that sounds unsnarky. I thought everyone learned about the polar representation of complex numbers.
Here is an even better discussion on the same topic, and the HN thread from last year: http://betterexplained.com/articles/a-visual-intuitive-guide... https://news.ycombinator.com/item?id=2712575 One great conclusion from this approach is how intuitive it becomes to understand the square root of i . I always thought you'd need another dimension to describe that, and another dimension for the square root of that unit,…
One more question; what is the "i"th root of i? Please try to use the angle metaphor :)
I like this question because it really works your intuition.
The basics: x^y means "grow at x, for y units of time". I see "2^3" as "grow at 2x for 3 units of time".
Having a base of i means your "growth" is a rotation at 90 degrees, no scaling. So i^(1/2) means a 45 degree rotation, i^3 means a 270 rotation, etc.
Raising this to the i power (or 1/i power, which is -i) means the growth that was originally purely rotational is now rotated. So instead of growing at i, you are growing at (i * 1/i = 1). So, we should expect a positive real number, greater than 1 since our growth is positive.
How long do we actually grow for? Well, the base of "i" is really e^(i* pi/2), which means "Start at 1.0 and rotate continuously for pi/2 seconds". We've now modified this to "Start at 1.0 and grow at 1.0 for pi/2 seconds", which is e^(pi/2).
So the answer is i^(1/i) = e^(pi/2) ~ 4.8
It's a bit tough with text-only, read the above article for more diagrams.