Read the "1" unit side as "move left 1 unit" and the "i" side as "move up i units", and the hypotrnuse is the net distance travelled.
The imaginary line is perpendicular to the real line, so "up i unit" is equivalent to "right 1 unit". The two movements cancel out giving a net distance of zero.
Yep. A vertical line segment above A with length 𝑖 is a horizontal line segment to the left that's 1 unit long. So, the diagram needs a "not to scale" caveat like a map projection, but there's nothing actually wrong with it, and the triangle's BC side is 0 units long.
Yeah. We were making a joke about the complex plane -- you could say that measuring the hypotenuse of a triangle is equivalent to measuring the distance between points |AB| and |AC|𝑖 on the complex plane. That definition actually makes quite a bit of sense, and I think by sheer coincidence it's possible to misunderstand how to do it and wind up with a way of looking at it where the hypotenuse of a right triangle with sides 1 and 𝑖 would work out to exactly 0. Which brings it back into concordance with OP's (also wrong) Pythagorean presentation of it.
It obviously doesn't really work that way, but it's hard to see necessarily anything wrong with it, which makes it a fun math thing.
Also, for your version, on a number line or Cartesian plane, the distance from -1 to 1 is 2, not 0
Yeah. I cheated. You have to either deliberately misunderstand how to measure vectors or else drop a minus sign for it to work my way.
(Or, from my previous example, you could just frame it as you're getting the hypotenuse by measuring between |AB| and -|AC|𝑖 instead of the way I framed it -- but that makes it more obvious that you're fishing for a particular answer.)
By far the most complicated part is the fact that the ratio of successive terms in the Fibonacci sequence approaches a specific number (which happens to be the golden ratio, which happens to be close to the ratio of km/mi).
What is the point of chili without meat? It would just be a bowl of beans and spices (or just a bowl of spices since beans really don't belong in chili).
You still have to prove that the integral exists, i.e. that from x=0 to x=∞ the integral is some number (not ∞ and not indeterminate). So it's not a total waste of time.
Edit: oh, the limits aren't ±∞ so if the function has no vertical asymptote in the range, it's a given.
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