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If we contruct two strainght lines as shown:enter image description here

Then join them such that to complete a triangle. enter image description here

It is taught that we can find infinity points on straight line. So there are infinity points on $DE$ and $BC$.

If we will join $A$ with $BC$ as shown:enter image description here We can find one point on $DE$ and corresponding point on $BC$. So point on $DE$ and $BC$ are same.

Hence, can we say that $\infty=\infty$, But why $\infty - \infty \neq0$

I'm not sure does this make any sense or not,your suggestions are appreciated.

mnulb
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    There is an honest, accurate, and extremely technical answer to your question. But in lieu of this, a heuristic reason is because you're not taking $\infty-\infty$. because a point is infinitely small, so it's more like you're taking $\frac{\infty}{\infty}-\frac{\infty}{\infty}$, and it's "well-known" that $\frac{\infty}{\infty}\neq\frac{\infty}{\infty}$. – Eric Stucky Jun 16 '16 at 06:05
  • What exactly is your argument? Are you saying that there is a bijection between points of DE and points of BC, but they are different lengths? – almagest Jun 16 '16 at 06:16
  • The problem is that «$\infty$» is not anything but just a colorful way of saying something, so «$\infty=\infty$» or «$\infty-\infty$» simpy do no tmean anything. – Mariano Suárez-Álvarez Jun 16 '16 at 06:24
  • You are not allowed to do computations with the infinity sign. Rather you should think of it as limits of points, so your limit of the difference of the number of points on the lines is equal to zero. – MrYouMath Jun 16 '16 at 06:29
  • @almagest I want to say there are same number of points on $DE$ and $BC$ and they are in infinite amount. There is one and only one point on $DE$ that is corresponding to $BC$'s one point. – mnulb Jun 16 '16 at 06:30
  • There is no difficulty with saying that $DE$ and $BC$ have the same number of points. That is true. So why do you say $\infty-\infty\ne0$? Do you mean that the length of $BC$ is greater than the length of $DE$? – almagest Jun 16 '16 at 06:34
  • @almagest $BC>DE$, conclusion would be that $\infty=\infty$ only. – mnulb Jun 16 '16 at 06:38

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Well first of all it is because $\infty$ is not a number, you can't do what you usually do with numbers but let's suppose here that it is, let's suppose $\infty$ is a number, the bigest number. Then you can imagine a finite constant $k$ added to $\infty$ which has to be $\infty$ also i.e. $k+\infty=\infty$. That then implies that $$\infty-\infty=k+\infty-\infty=k$$ This would be true if $\infty$ was a defined number in the first place which it isn't but we assumed it was here and we found that $\infty-\infty=k$, which remember $k$ can be any number you want.

We say then $\infty-\infty$ is an undefined expression. Not because it is infinite, actually, not mainly for that reason. It is actually due to the fact that $\infty$ is not a defined number like $2$ or $\pi$ or other numbers like them.

We could define an infinite number. Consider the product of all natural numbers from 1 and give it a name, say $I$. Then $I-I$ really equals $0$ because I really is a mathematical object in this case and really equals the same thing. It would be surely weird to work with such infinite numbers, but it already has been done.

So say your last line is twice the length of the first, you say "well there are twice as many points in that second line", but consider this: it is well know that there is the same "infinity" of natural numbers than the "infinity" of even numbers, is because there exists a one-to-one correspondance between the sets, even if you may think, at first sight, that there are twice as many naturals than there are even numbers. Infinity is tricky and is treated differently whether you are talking about an amount, in sets of numbers or points for example, or more like a number.

abcabc123
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Take this example. Find any point between $(1,\infty)$ , invert it and you will get a point in $(0,1)$. Since there are infinitely many points you can choose , according to you it must mean $\infty=\infty$ i.e. $(0,1)=(1,\infty)$?

Qwerty
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  • If by $\infty$ we mean "the cardinality of the continuum" (which is a reasonable interpretation, since he is talking about points), then your argument is valid, yes. (well, up to the point where you say that equality of numbers implies an equality of sets— which his argument does not do anywhere, fwiw— but just stick a cardinality marker in front of the intervals and it's fine.) – Eric Stucky Jun 16 '16 at 06:09
  • @EricStucky As far as i thought , the OP did not mean "cardinality" when he said that $DE$ and $BC$ are same. (He didnot say $#$ points in $DE$ and $BC$ are same.) – Qwerty Jun 16 '16 at 06:13
  • My parenthetical above was meant to reference OP's sentence "We can find one point on $DE$ and corresponding point on $BC$. So point on $DE$ and $BC$ are same". Perhaps I was reading this more generously than you were? it's possible that the OP thinks that the two points are literally the same, in which case they're just wrong. I assumed they meant "the amount of points" in some vague sense, in which case "cardinality" still strikes me as closest to what they were getting at. – Eric Stucky Jun 16 '16 at 06:16
  • @EricStucky Yup. I see you perspective.. You were indeed reading the question more generously than required I think..I think the OP has collected a wrong concept and needs to be corrected(Comments from others also hint that).That's why I gave an example to highlight his/her fallacy. – Qwerty Jun 16 '16 at 06:33
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the problem is that when we define point we consider point to be dimensionless but line is comprised of points , so it should be dimensionless as well but that's not the case . To go around the problem consider an $\epsilon$ value that is the minimum distance between two points , now you will realize for that $\epsilon$ value in $DE$ the distance between two points in $BC$ is more than $\epsilon$ thus if we place points optimally we can place more points than that on $DE$ , and ration of those points will tend to ratio of their lengths

avz2611
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