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on 1.2) is 1 divided by zero not undefined instead of infinite?
Earlier quoted context omitted.
The multiplicative inverse, or reciprocal, or the inverse of a number, is a function acting on a number, 1/x, or x^-1.
Isn't this a case of multiplication being such a "well-known" function that the "inverse" defaults to "multiplicative inverse"?
If we're talking about trig functions, inverse sine or arcsine is very different than just the reciprocal of the sine.
If we're talking about images, inverse usually means "rotate 180 degrees" or "color inversion".
The answer is “it depends on how you defined the operation that acts on the circle”
As long as, C*Cinv = I, where C is the circle, Cinv is the inverse of circle, and I the identity. You're right. C, I, and * are entirely up in the air.
If F is defined as `ra•rx`, then `ri == 1`, and inverse will be `rx = 1/ra`.
If F is defined as `ra + rx`, then `ri == 0`, and inverse will be `rx = 0 - ra`, where negative radius means hole.
If F is defined as `ra²•rx²`, then `ri == 1²`, and inverse will be `rx = sqrt(1/ra²)`.
If F is defined as `ra² + rx²`, then `ri == 0²`, and inverse will be `rx = sqrt(0 - ra²)`.
And so on.
Earlier quoted context omitted.
Isn't this a case of multiplication being such a "well-known" function that the "inverse" defaults to "multiplicative inverse"?
No, inverse means reciprocal only when the context is a number or algebraic variable. If we're talking about trig functions, inverse sine or arcsine is very different than just the reciprocal of the sine. If we're talking about images, inverse usually means "rotate 180 degrees" or "color inversion".
If my function to generate a circle is a simple for loop 0 to 2 PI.
Then the inverse of that maps each point on the circle back to a line with points between 0 and 2 PI.
The answer is “it depends on how you defined the operation that acts on the circle”
In polar coordinates a circle can be defined by all points where r = a. The inverse would be all points where r doesn't equal a.
* The inverse of a geometric shape makes no sense. We only inverse operations.
* aa^-1 = 1 only if you consider the multiplication over reals.
* 1/0 is not equal to infinity.
Because the article is interesting but some people might be put off by the first few sentences, I suggest to had a disclaimer that this article lean on edutainment to the detriment of rigorous mathematics.
Maybe this article is way over my head, but is it not really simple? If my function to generate a circle is a simple for loop 0 to 2 PI. Then the inverse of that maps each point on the circle back to a line with points between 0 and 2 PI.
This article is instead talking about the inverse per the identity a * 1/a = 1.