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I am learning the number theory. I am not a native English speaker. I encountered the word commensurable from time to time. Such as:

the Pythagoreans discovered when they found that the diagonal of certain geometric fgures may be incommensurable with their sides.

I guess I need to understand the motivation/origin/construction of the word commensurable to get its real meaning.

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As I googled, there can be 3 meanings as below. I think the 1 is reasonable based on the word's origin. But how come it leads to ratio in 2 and especially, ratio of integers in 3??? What's the mind flow of a native English speaker for this?

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    C'mon, guy. All your previous questions on this site were closed for lack of research. Show some respect for the site. Indicate your research, Put your research into your question, so we don't duplicate it and we can help you better... Sep 29, 2018 at 4:18
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    Delve into history after you know it's meaning. Otherwise you'll go nuts.
    – Aethelbald
    Sep 29, 2018 at 7:08
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    The real meaning is #3, because the context is mathematics. Sep 29, 2018 at 8:36
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    He thinks that this guy really likes to use big words.
    – Hot Licks
    Oct 29, 2018 at 11:24
  • @HotLicks Yes, I don't prefer to big words. I like words that can make things clear. Oct 30, 2018 at 1:43

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To answer the question, which is about what a native person thinks, not about what it means:

In the context of the history of mathematics or of science, I understand the term. In any other context I would think it meant that two things can be measured against each other. For example the weights of iron and copper are commensurable but the weights of iron and opinion are not. But I might also think the speaker was a bit pompous.

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  • Thanks. Your explanation and analogy is very interesting. Oct 30, 2018 at 1:28
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It might help to imagine constructing a right-angle equilateral triangle out of lego blocks. Say 10 blocks each for the equal sides. Can you make a diagonal that fits exactly? No. This is the substance of the ancients' problem with commensurability.

Two non-zero real numbers a and b are said to be commensurable if their ratio a/b is a rational number; otherwise a and b are called incommensurable. Recall that a rational number is one that is equivalent to the ratio of two integers.

So consider the square root of 2, or SQRT(2), which is not an integer.

Now consider a right-angled isosceles triangle where the two equal sides have length 1 and the hypotenuse is SQRT(2).

The ratio of the hypotenuse to one of the sides is SQRT(2)/1 which is cannot be described as the ratio of two integers. Therefore the sides are incommensurable.

BTW, very few native English speakers have "commensurable" in their primary or secondary vocabularies. It's really for the mathematically inclined.

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