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What will be the difference, if we calculate difference in lengths of two lines which are along x axis, one of which is a line starting from 1 and ending at infinity and other line is starting from 7 and ending at infinity? To my opinion It has to be exactly 6 units.Why is the answer six? Because we can not actually measure the infinity but we can still comment on the difference between two lines having infinite length. So is it like we can not measure lines with infinite length, but can comment about their comparisons if they have at least one end which is finite?

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Before ask "what is the difference", perhaps you should first find a rigorous definition of "difference". The same concept can have multiple definitions in mathematics, each leading to different answers. This is why one should always first nail down definitions, as well as explore various definitions. –  rghthndsd Jul 20 '13 at 17:01
    
I mean the difference which means minus sign (-) in algebra here. –  Deepeshkumar Jul 20 '13 at 17:05
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@Deepshkumar: But the minus sign takes two numbers and gives you another number. It doesn't know anything about these crazy lines you're drawing! –  Sharkos Jul 20 '13 at 17:06
    
What I meant was if we measure the difference in lengths of two lines along positive x axis , one of which is a line from 1 to infinity and other is from 7 to infinity the difference has to he 6 units exactly. Am I correct because many people I asked strongly said that it is infinity. What is a correct answer? –  Deepeshkumar Jul 20 '13 at 17:49
    
@Deepshkumar: You have the same exact problem! What is the definition of the length of a line with (possibly) no bound on it? Until you decide what this means, you can't ask your question. Your question depends on the meaning of length. –  rghthndsd Jul 20 '13 at 17:55
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I think you mean the following. Let $$A=[1,\infty)$$ and $$B=[7,\infty)$$ Then $$A\smallsetminus B=[1,7)$$

That is perfectly fine, but maybe a better wording was necessary!

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wouldn't it be $[1,7)$ –  DepeHb Jul 20 '13 at 17:15
    
@DepeHb Sure. ${}{}{}{}$ –  Pedro Tamaroff Jul 20 '13 at 17:20
    
Yes Peter I agree with you. Is there any book that explains infinity concept and its impact on mathematics? I have not read much than a highschool grade maths book in India.Want to improve my knowledge. I adore your mathematical explaination ability and analytical ability. Quite awesome. –  Deepeshkumar Jul 21 '13 at 6:49
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The most natural answer is 6, but this isn't at all mathematical yet.

Instead, take two lines $A,B$ and define the difference line $A \Delta B$ to be the bits of line in one of $A,B$ but not in the other. (This is called the symmetric difference.) Then you can ask how much all those bits of line add up to if you stick them together. Here, you get six.


The essential problem faced in getting an answer to this question is simple: you're speaking a different language to us. And I don't mean French instead of English, Welsh instead of Maori, Hindi instead of Urdu, or anything like that. I mean you are not speaking in a mathematical language.

Almost everything a mathematician writes - unless they add words like "roughly" or "philosophically" or "you can think of this like" to show they are being vague - is in a language which could be translated directly into symbolic logic, with every single notion being rigorously defined.

You waltz in, with a normal sane mind, and ask what sounds to you like a reasonable question, "how big is the difference between these two lines" or whatever. Mathematicians react badly to this, out of training and discipline and, yes, some impatience, mainly because we've seen it all before (that is, loose imprecise wording leading to meaningless-ish questions with no practical answer, often with words like infinity). (It's tough being different and weird!) The point is there is and can be no a priori right answer to your question. What there can be is an explanation for why we might define various possible things to be the right answer. But you didn't know this when you asked.

Mathematicians look at that and they know you don't have a clear enough notion of "difference" in mind, or of size, and will (based on this thread P:) either (a) ignore this and give an answer which assumes some specific interpretation you've not worked with before, (b) answer the wrong question entirely, (c) be turned off and go away and be angry, (d) ask you what meant and try to help you figure it out or give space for you to do so, or (e) suggest an interpretation and explain how it might be used. The people who put the question on hold were probably mainly in (d).

I hope this helps you to see how mathematicians answer questions like this, and adds something to your understanding of the answers you got.

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You are the only person who got me right. I have ellaborated my question by editing it. They have put it on hold. My mother tounge is not english does that mean I loose a right to ask? –  Deepeshkumar Jul 20 '13 at 18:19
    
Let me assure you that there is no conspiracy! The problem is... actually, I'll add this to the answer. Hang on. –  Sharkos Jul 20 '13 at 19:47
    
@Deepeshkumar I think I got you right too. Since in this case $A\subseteq B$ we have $A\setminus B=\varnothing$, thus we have $A\triangle B=B\setminus A$. –  Pedro Tamaroff Jul 20 '13 at 20:09
    
I agree that @PeterTamaroff is another category (e) for instance! (: –  Sharkos Jul 20 '13 at 20:28
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For the exact same reasoning (starting at two instead), one can conclude that $\infty - \infty = 1$; likewise, you could extend this to every real number.

This is why one cannot make a definition for $\infty - \infty$ that is consistent with all the usual properties of subtraction and addition. In particular, it simply demonstrates that $\infty$ is not a real number in the usual sense, and so trying to do arithmetic with infinity requires very careful considerations.

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If I understand you right, your question is:

How much is $(\infty -1) - (\infty - 7)$?

What you did is $$a := \lim_{x \to \infty} (x-1), \quad b := \lim_{x \to \infty} (x-7),$$ so $$a = \infty - 1, \quad b = \infty - 7.$$ Then, you conclude: $$a - b = \lim_{x \to \infty} (x-1) - \lim_{x \to \infty} (x-7) = \lim_{x \to \infty} ((x-1) - (x-7)) = 6.$$

Now, consider this: $$a := \lim_{x \to \infty} (2x-1), \quad b := \lim_{x \to \infty} (x-7)$$ Again, $$a = 2\infty - 1 = \infty - 1, \quad b = \infty - 7.$$ But, $$a - b = \lim_{x \to \infty} (2x-1) - \lim_{x \to \infty} (x-7) = \lim_{x \to \infty} ((2x-1) - (x-7)) = \lim_{x \to \infty} (x-6) = \infty - 6.$$ Whoops.

The problem is that you treat $\infty$ like a number, which it is not. There is infinitely (pun intended) many ways to get infinity, and they will give you as many different answers to your question.

You can even get it upside down, i.e., $(\infty -1) - (\infty - 7) < 0$. Just put $$a := \lim_{x \to \infty} (x/2-1), \quad b := \lim_{x \to \infty} (x-7).$$

The correct answer to your question is: "undefined, as long as you don't define your infinities (i.e., by the limits, like I did)".

Line lengths

Since this was written, there was some clarification that you are talking about line lengths. The answer is still: "undefined".

I'll use other peoples' example: $$A := [1, \infty \rangle, \quad B := [7, \infty \rangle.$$ This looks like $A$ is longer by $6$, but what do you get if you move (without changing the length!) $B$, let say, by $10$ to the left? You get $$B' := [-3, \infty \rangle,$$ which looks like it is longer than $A$ by $4$. But, I repeat, we have just moved $B$, we didn't change its length!

So, again, no, you cannot do it like this.

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I don't think that is the question being asked. –  Pedro Tamaroff Jul 20 '13 at 17:18
    
@PeterTamaroff Might be. I drew my conclusion from his comment "I mean the difference which means minus sign (-) in algebra here". –  Vedran Šego Jul 20 '13 at 17:48
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If you want to consider this as problem of comparing 2 numbers on $$ \overline {\Bbb R} $$ then $$ (\infty - 1) - (\infty - 7) = \infty - \infty = undefined $$ If you want to consider this as a problem of comparing two lines: $$ A=[1, \infty) $$ $$ B=[7, \infty) $$

then $$ |A| = \infty $$ and $$ |B| = \infty $$ Hence $$ |A| - |B| = \infty - \infty = undefined $$

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What happened to 1 and 7? Though infinity is uncountable, u can subtract them and the substraction is always zero. So the answer is 6. –  Deepeshkumar Jul 20 '13 at 19:13
    
No you can't, because + isn't associative on $$ \overline {\Bbb R} $$. –  daniel Azuelos Jul 20 '13 at 23:03
    
@danielAzuelos Actually, $|A|=|B|$. That is pretty well defined. –  Pedro Tamaroff Jul 21 '13 at 0:01
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