I understand that irrational numbers are categorized by being numbers that continue infinitely. I guess my question is more precisely, are the infinite sets of irrational numbers considered rational numbers themselves? (Ex: {.30,.31,.32,.33,.34,.35,. . . . })
2026-03-30 09:43:43.1774863823
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Are Irrational Numbers also Rational Numbers?
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No, sets of irrationals are not rational numbers, although provocatively, there is a construction of the reals called Dedekind cuts, in which every real number (of which the overwhelming majority are irrational) is identified by a particular partition of the rationals into two subsets, with one subset $L$ containing all numbers below, and the other subset $R$ contains all numbers above, and $L$ not containing its supremum, or lowest upper bound. (Equivalently, neither $L$ nor $R$ has a greatest element.)
Dedekind cuts have other applications, but this is the best known one.
To be sure, one could reverse the Dedekind cut and identify each rational number uniquely with the subset of irrational numbers that are less than that rational number. However, I would hesitate to call that subset an actual rational number, since no one constructs rational numbers that way; it would be begging the question (circular reasoning).
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The rational numbers, in decimal representation, have a finite or a infinite and periodic sequence of decimals (indeed we can say that a number with finite decimals have infinite zeros after, what is a periodic sequence after all).
In the other hand irrational numbers, in decimal representation, have an infinite sequence of decimals that is not periodic. This is the source of your confusion.
Just as a little curiosity (that would you surely surprise) is that
$$1.000000...=0.999999...$$
what imply that the decimal representation of numbers is not unique in many cases, by example
$$3.47=3.4699999...$$
No.
Rational numbers are numbers that can be written as a fraction $\frac ab$ with $a\in\mathbb{Z}$ and $b\in\mathbb{N}$. Irrational numbers are defined to be the opposite, numbers that can't be written that way.
To expand on this; it's (in my opinion) a bad custom to view numbers as "infinite digits behind a comma". Often, numbers don't even have a unique such representation.