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Can irrational numbers be thought of as ratios of infinite quantities?

Arseland · 2015-11-24 16:51:28

The classic proof that root 2 is irrational is to assume it can be represented as a ratio a/b and then show that a and b must have always the common factor of 2 which means that it can't be simplified to the point that a and b are mutally prime. But what if a and b were infinite? You could keep on dividing them by 2 and…

24-11-2015 05:51PM

#1

Arseland

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The classic proof that root 2 is irrational is to assume it can be represented as a ratio a/b and then show that a and b must have always the common factor of 2 which means that it can't be simplified to the point that a and b are mutally prime.

But what if a and b were infinite? You could keep on dividing them by 2 and never reach bottom.

Can this be made rigorous or is it just a case of infinity/infinity = anything?

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#2 25-11-2015 05:30AM

dlouth15

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The definition says that a number is rational if it can be expressed as the ratio of two integers. However infinity is not an integer and therefore can't be used to express a rational. a and b must both be finite.

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