Why are "irrational" and "transcendental" so commonly used to describe numbers

[rumborak]

(sorry, the thread title got mangled. It should be "why are irrational and transcendental so commonly used to describe numbers")

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?

 

[scottdave]

If a number is rational, then you can store its exact value (or a way to compute its exact value to desired precision). For example 1/2, you can store a 1 and 2 then just divide them and get a nice 0.5 decimal. Even repeating decimals such as 1/11 or 1/7 can be divided and you will get a decimal which goes forever, but can be handled.

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 x² - 2 = 0. Even the golden ratio is (1/2) + sqrt(5)/2, and is a solution of a polynomial. Transendental numbers like e and pi cannot be solutions to a polynomial such as this. You may find this video helpful.

 

[fresh_42]

It is no hierarchy, it is a property. Numbers which fulfill a polynomial equation are called algebraic, and numbers which do not fulfill such an equation as e.g. $\pi$ over $\mathbb{Q}$ are called transcendental. It is not obvious whether a number is transcendental or not. It could simply be the case that we haven't found an appropriate equation yet and for many numbers, it is still not clear whether they are algebraic or transcendental. The proof that $\pi$ is transcendental dates back to 1882 (F. v. Lindemann). This is not that long ago considered since when we deal with $\pi$.

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.

 

[mathman]

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

 

[SSequence]

...
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.

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).
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).

 

[fresh_42]

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).

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.