How $\infty=\infty$.

Question

If we contruct two strainght lines as shown:

Then join them such that to complete a triangle.

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: 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.

Answer 1

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. $(0,1)=(1,\infty)$?

Answer 2

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

Answer 3

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.