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What is the meaning, when it is the irrationality of a constant?

I understand that irrational means can't be represent as a fraction.

Toy
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  • Compare the phrases "The irrationality of $e$" and "The divinity of God." (Ignoring the religious debate for the time being, just using an example of a similar word in use). $e$ is irrational. God is divine. When we refer to the divinity of God, we refer to those qualities which make God divine. Similarly, when we refer to the irrationality of $e$, we refer to those qualities which make $e$ irrational. In effect, "irrationality" and "divinity" are both nouns while "irrational" and "divine" are both adjectives. – JMoravitz Jun 18 '18 at 02:50
  • As for what does "irrational" mean, a real number is considered irrational if and only if there is no way for it to be represented as a ratio of two integers $\frac{p}{q}$ with $q\neq 0$. (An irrational number could still appear in a ratio or as a ratio so long as those numbers in the ratio could be irrational themselves, for example $e=\frac{e}{1}$) – JMoravitz Jun 18 '18 at 02:52

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We can define constants in many ways. Sometimes it is not clear whether the constant is rational or irrational. An example would be $$c=\sum_{i=1}^\infty \frac 1{i^2}$$
Once I prove convergence, this is a fine definition of $c$. From this definition, it is not obvious whether $c$ is rational or irrational. As there are so many more irrational numbers than rational numbers, we would be prone to guess that $c$ is irrational because we don't see any reason it should be rational. In this case we know it is $\frac {\pi^2}6$ which is irrational.

Ross Millikan
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  • How can there be more irrational than rational? This seem to contradict the logic of the mind. – Toy Jun 18 '18 at 06:39
  • If the above statement is true than most of these constants, such as Catalan, Euler, odd zeta constants are highly irrational, but mathematicians still try to prove it. Why bother? – Toy Jun 18 '18 at 06:43
  • We can show there are only countably many rationals. Cantor's pairing function is one route. Cantor's diagonal proof shows there are more reals, so there are more irrationals than rationals. One can make precise that the rationals are measure zero in the reals, so the chance a "random" number is rational is zero. It is an interesting fact that a given constant is irrational, but often it is difficult to prove. I don't know what you mean by highly irrational. – Ross Millikan Jun 18 '18 at 13:49