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A strange behavior of numbers

Let there be m real numbers greater than 1, be called a set S. If we reciprocate every element in set S, all the numbers will lie between 0 to 1. Hence, number of real numbers between 0 to 1 is m.

1.Now, let us consider the number of real number between 1 to 2 be n. It is obvious that m>n.

2.If we add 1 to all the elements in set S we get a new set of real number (be called a set Q) which has all it's elements between 1 to 2 and the number of elements is m (as we are adding 1 to all the elements in set S).

In para (1) we are saying number of that the number of real numbers between 1 to 2 is n (which is less than m) and in para (2) we are saying that number of real number between 1 to 2 is m.

Why are these two contradicting each other?

And how can there be equal number of real number between 0 to 1 and 1 to infinity?

Note by Rahul Dandwate
4 years, 2 months ago

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This is opening a really big topic, and one which raises some very difficult ideas. We say that two sets have the same number of elements if we can find a one-to-one correspondence between their elements. A one-to-one correspondence between sets \(A\) and \(B\) is, basically, a function from \(A\) to \(B\) such that

  • distinct elements of \(A\) are mapped to distinct elements of \(B\),

  • every element of \(B\) is the image of some element of \(A\).

This reasonable suggestion has apparently weird consequences. For example, there are as many positive integers as there are even positive integers, since the correspondence \(n \mapsto 2n\) puts the positive integers and the even positive integers into one-to-one correspondence. It "gets worse", since there are as many rational numbers as there are integers!.

It turns out that there are more real numbers than there are integers. However, there are the same number of real numbers between \(0\) and \(1\) as there are between \(0\) and \(2\), with the map \(x \mapsto 2x\) being the one-to-one correspondence we need.

The function \(f(x) = \tfrac{x}{1+x}\) provides a one-to-one correspondence between the positive real numbers and the positive real numbers less than \(1\), so these two sets have the same number of elements.

If you want to play with some of the ideas relating to infinity, try searching for information on Hilbert's Hotel on the web.

Mark Hennings - 4 years, 2 months ago

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"It is obvious that m > n." Is it?

Michael Tang - 4 years, 2 months ago

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It should be as a whole is always greater than part of it.

Rahul Dandwate - 4 years, 2 months ago

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That's the part where your argument breaks down. With infinities, a whole is not necessarily greater than the part of it, although it's never less. In this particular case, turns out the whole is the same as the part.

Ivan Koswara - 4 years, 2 months ago

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Your argument fails at the first condition itself. \(m\) is not necessarily greater than \(n\); a one-to-one correspondence could be established.

Sometimes it is not so easy to imagine this by plainly looking at sets; a picture is sometimes necessary. You'll be interested by the bottom portion of this diagram. Imagine each point on the semicircle representing a real value between 0 and 1 (with 0 and 1 represented by the endpoints of the diameter of the semicircle). Each point on the line represents a real number (with the left side going towards negative infinity and the right side going towards positive infinity). If one draws any line from the center of the semicircle to the line below it, exactly two points or no points are encountered for every possible line drawn. Thus, a one-to-one correspondence can be established between the real numbers in any finite range and the real numbers in an unbounded range.

I leave the exercise to you to try to find the one-to-one correspondence between the sets described above (I highly suggest a geometrical approach.)

Tanishq Aggarwal - 4 years, 2 months ago

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