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I have been considering the properties of infinity and applications to various areas of maths and was hoping to get some opinions of more seasoned mathematicians than myself. One geometric representation of infinity could be that of a vertical line which would theoretically have an infinite gradient if we were to approach the thought from the perspective of limits. However that would imply that $$\frac{1}{0}=\infty$$ If this were true it could be used to prove a whole bunch of things that simply aren't true e.g $$1=0\cdot\infty=0$$ What is the flaw in the geometric representation of an infinite gradient? Any thoughts on the matter are greatly appreciated!

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  • $\begingroup$ You may interested in the projective space. $\endgroup$ – Hanul Jeon May 12 '15 at 14:02
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You're talking specifically about the projective compactification of the number line, and the issue is that in that setting it doesn't make sense to multiply numbers together. Or, more specifically, you just cannot multiply $0 = [0:1]$ by $\infty = [1:0]$, since this would result in $[0:0]$ which does not represent a point on $\mathbb{P}^1$ at all.

Read more: http://en.wikipedia.org/wiki/Projective_line

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You can extend the real numbers like that, as an ordered set, but you cannot extend the basic algebraic operations to that extended set in a way that allows you to do things like cancel terms from both sides of an equation. You need to be able to do those types of algebraic operations to prove the things that result in absurdities. To do those kinds of operations you need an honest-to-god ring without zero divisors. The area of math you should study to learn this stuff is Abstract Algebra.

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