- #1

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Is this simply out of the most common ways of how one would try to describe a number? (e.g. first try ratios, then polynomials) Or is there a deeper reason for this hierarchy of classifications?

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- Thread starter rumborak
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- #1

- 706

- 154

Is this simply out of the most common ways of how one would try to describe a number? (e.g. first try ratios, then polynomials) Or is there a deeper reason for this hierarchy of classifications?

- #2

- 1,814

- 778

Irrational numbers, such as square root of 2, cannot be calculated this way. The digits never repeat. But square root of 2 can be a solution to a polynomial with integer coefficients. It is a solution to x

- #3

fresh_42

Mentor

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Irrational simply refers to numbers which are not elements of ##\mathbb{Q}## - not rational. So ##\pi## as well as ##\sqrt{7}## are irrational. But while ##\pi## is transcendental with respect to ##\mathbb{Q}##, ##\sqrt{7}## is algebraic with respect to ##\mathbb{Q}##. The fact that indeed ##\pi \notin \mathbb{Q}## is known since 1767. So even this "obvious" result took its time before it could have been considered certain.

- #4

mathman

Science Advisor

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Added note: Algebraic numbers are countable, implying most numbers are transcendental.

- #5

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For many years I used to think that the value 22/7 was actually an exact one. Though perhaps, just on surface, one might have considered it a little odd based on circumference of a circle being increasingly approximated through square roots (but still, I guess one can't be sure unless one has ruled out a rational number as a limiting value of a sequence involving square roots).....

The fact that indeed ##\pi \notin \mathbb{Q}## is known since 1767. So even this "obvious" result took its time before it could have been considered certain.

But probably, when I first heard about someone remembering "digits" of pi, it occurred to me that it wouldn't make much sense if it was a rational number (specifically 22/7).

- #6

fresh_42

Mentor

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Digits are always merely a certain representation. The question why it was interesting to know, whether ##\pi ## is rational or in this case transcendental was another. It is one of the three classical problems, namely whether a circle could be squared, such that they both have the same area and only compass and ruler were allowed to use. The fact that ##\pi ## is transcendent answers this question (negatively). So people where interested in this question for centuries and long before we started to use digits.But probably, when I first heard about someone remembering "digits" of pi, it occurred to me that it wouldn't make much sense if it was a rational number (specifically 22/7).

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