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I would like someone to give me a definition of what irrationality measure is, I have stumbled over several definitions which may be equivalent but as I lack understanding I cant see this correlation. So I am looking for a definition and also a brief explanation of what it really tells us, is it something that tells us "how irrational" a number is?

I have decided to study this subject a little more since it was part of the answer to my question in this thread: Is there a proof for the following series to diverge/converge? (I dont know if my tags are ok)

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  • $\begingroup$ Can you post some of the proposed definitions so we can evaluate them? $\endgroup$ – Hew Wolff Jul 3 '14 at 21:44
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Since no one has given an answer, I'll offer some suggestions. However, this isn't something I know all that much about, despite the fact that I somehow managed to answer a recent question on the topic: Does every normal number have irrationality measure $2$?.

I don't know your background, so I'll start low and end high.

A really nice and elementary place to begin reading about underlying ideas is with Chapters 6 and 7 in Ivan Niven's Numbers: Rational and Irrational. After this, maybe look over Chapter XI in Hardy/Wright's An Introduction to the Theory of Numbers. This should get you far enough to have a reasonable sense of what the topic is about, and where you go next depends a lot on what your mathematical background is.

Some advanced references that seem worth looking at include the following:

Approximation by Algebraic Numbers by Yann Bugeaud

Transcendental Number Theory by Alan Baker (Chapter 8)

Making Transcendence Transparent by Burger/Tubbs (Chapter 6).

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