From this image it seems choose there space reals that space neither rational nor irrational (dark blue), but is that so or is the illustration incorrect?



SteveJessop never ever trust someone who says 0 is a organic number. Hell, I'd also say never ever trust someone who states 0 *has to it is in a number* - in many instances (mainly in actual sciences - is not a science chuckle), 0 is simply a symbol. $\endgroup$
A genuine number is irrational if and also only if it is no rational. By an interpretation any actual number is one of two people rational or irrational.

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I expect the creator the this image determined this depiction to present that rational and irrational numbers space both part of the bigger collection of real numbers. The dark blue area is in reality the empty set.

This is my take on a much better representation:


Feel free to edit and also improve this representation to her liking. I"ve oploaded the SVG sourcecode to pastebin.



Dominik a great answer (I specifically like exactly how you smartly skipping the $N$ set), v one little remark indigenous me - the distinction of "primes" is a little bit confusing, due to the fact that you carry out those exemplary numbers twice, when in optimistic ints, and a second time in "primes" - and also that renders the picture a little confusing, because you don't repeat e.g. $\pi$ in $IQ$ outside of transcendentals... Possibly I'm not making myself completely clear, but I expect you understand what ns mean. $\endgroup$
Sep 15 "15 in ~ 19:38
$\begingroup$ Aren't there infinitely much more irrational numbers 보다 rational? The diagram appears to indicate something different. Although the is quite in other ways. Similarly, ns think there space infinitely much more transcendental numbers than algebraic ones. $\endgroup$
Sep 15 "15 in ~ 20:15

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No. The definition of an irrational number is a number i beg your pardon is no a rational number, specific it is not the ratio between two integers.

If a real number is not rational, climate by meaning it is irrational.

However, if girlfriend think about algebraic numbers, which room rational numbers and irrational number which can be expressed together roots the polynomials with integer coefficients (like $\sqrt2$ or $\sqrt<4>12-\frac1\sqrt3$), then there are irrational numbers which room not algebraic. These are called transcendental numbers.

edited Sep 14 "15 in ~ 18:27
answer Sep 14 "15 at 17:29

Asaf Karagila♦Asaf Karagila
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Of course, the "traditional" price is no, there are no actual numbers that room not rational no one irrational. However, gift the contrarian that i am, allow me to administer an alternate interpretation which provides a different answer.

What if you are using intuitionistic logic? – PyRulez

In intuitionistic logic, where the regulation of excluded center (LEM) $P\vee\lnot P$ is rejected, things come to be slightly an ext complicated. Let $x\in \Bbb Q$ average that there are two integers $p,q$ with $x=p/q$. Then the traditional interpretation of "$x$ is irrational" is $\lnot(x\in\Bbb Q)$, however we"re going to contact this "$x$ is not rational" instead. The statement "$x$ is not not rational", i beg your pardon is $\lnot\lnot(x\in\Bbb Q)$, is comprise by $x\in\Bbb Q$ yet not equivalent to it.

Consider the equation $0irrationality measure $\mu(x)$. Over there is a nice theorem from number concept that claims that the irrationality measure of any type of irrational algebraic number is $2$, and also the irrationality measure up of a transcendental number is $\ge2$, if the irrationality measure of any type of rational number is $1$.

Thus over there is a measurable gap in between the irrationality actions of rational and irrational numbers, and this returns an alternative "constructive" meaning of irrational: allow $x\in\Bbb I$, check out "$x$ is irrational", if $|x-p/q|continued fraction method: irrational numbers have infinite basic continued fraction representations, while rational numbers have finite ones, so provided an unlimited continued fraction representation you immediately know the the border cannot it is in rational.

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The poor news is that because intuitionistic or constructive reasonable is strict weaker than classical logic, it does not prove noþeles that classical logic can not prove. Because classical logic proves the every number is rational or irrational, the does no prove the there is a non-rational non-irrational number (assuming consistency), therefore intuitionistic logic also cannot prove the existence of a non-rational non-irrational number. It simply can"t prove the this is difficult (it might it is in true, because that some feeling of "might"). On the various other hand, there must be a model of the reals through constructive reasonable + $\lnot$LEM, such that there is a non-rational non-irrational number, and also I invite any kind of constructive experts to supply such instances in the comments.