transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
An irrational number is of course a number that is not rational. As such, the concept is perhaps uninteresting. However, the term ‘irrational number’ is often used for an irrational real number; in this case, it is interesting to consider such numbers for two reasons:
Of course, there are also various theorems about general classes of numbers that distinguish rational from irrational numbers.
An affine function on the real numbers consists of a function and real number coefficients and such that for all real numbers , . The constant function at zero is an affine function where and . A real number is irrational if for all affine functions with integer coefficients, if , then . This is equivalent to saying that a real number is irrational if for all rational numbers , .
Alternatively, a real number is irrational if given any rational number (thought of as a real number), the absolute value is positive.
These two definitions are equivalent in classical mathematics. However, these two definitions no longer coincide in constructive mathematics; the former definition of irrational number is called weakly irrational while the latter definition is called strongly irrational or strictly irrational. Strongly irrational numbers are most commonly used in constructive mathematics, since it uses the apartness relation or strict order relation of the real numbers, which, unlike equality, is what is detected of the real numbers in constructive mathematics.
The set of irrational real numbers (a subset of the set of real numbers) is variously denoted , , or (in various fonts). The and stand for ‘irrational’, while the stands for ‘Baire’ (see the next paragraph). Here we will use .
We may give a topology as a subspace of the real line . With this topology, is sometimes called Baire space; however, one uses a different uniform structure. (This should be distinguished from the sense of Baire space as a space to which the Baire category theorem applies; however, is an example of such a space.)
There is another definition of irrational number, common in the prealgebra and high school algebra literature, which directly defines the irrational numbers in terms of base 10 infinite radix expansions. (see prealgebra real number). This can be done in every base greater than 1:
Let the natural number denote the base of the radix expansion, denote the half-open interval in the natural numbers of all natural numbers less than . Base infinite radix expansions are elements of , with the idea that each pair consists of an integer and a sequence of digits in the base infinite radix expansion. The series
can be shown to be a Cauchy sequence.
The set of repeating base infinite radix expansion is the subset of such that for pairs in the subset, there exist natural number and positive natural number such that the sequence factors through the cyclic group .
Two base infinite radix expansions and are said to be apart from each other if or there exists a natural number such that
A base infinite radix expansion is strictly non-repeating if it is apart from every repeating base infinite radix expansion. An irrational number is a strictly non-repeating base infinite radix expansion.
In constructive mathematics, not every strictly irrational number is the limit of the series
given by a non-repeating base infinite radix expansion. However, it is still the case that strictly irrational numbers which are also Cauchy real numbers are interdefinable with strictly non-repeating base infinite radix expansions.
Let be an Archimedean integral domain with the integers being a integral subdomain of .
An element is irrational if for all and , and .
The set of irrational numbers in is defined as
Let be an integral domain with a p-adic norm for a prime number , with the integers being a integral subdomain of .
An element is irrational if for all and , and .
The set of irrational numbers in is defined as
The followers of Pythagoras believed that ‘All is number’, meaning what we now call (positive) natural numbers. In geometry, this meant that any two lengths (or other geometric magnitudes) and are commensurable? in the sense that there exists a unit length such that and for some natural numbers and . Identifying the ratios of geometric magnitudes with (positive) real numbers, this becomes the claim that every real number is rational. The discovery that this is false is also attributed to the Pythagoreans (but the legends of punishment for this secret date from several hundred years later). Greek mathematicians developed further the theory of irrational numbers, up to the general theory of magnitudes (which we may now regard as a theory of positive real numbers) attributed to Eudoxus in Book X of Euclid's Elements.
Mathematicians coming from the cultures of the Islamic Golden Age (particularly Abu Kamil?) were the first to treat irrational numbers algebraically as numbers (rather than geometrically as ratios of magnitudes); they applied the algebra of Al-Khwarizmi? to square roots, cube roots, etc. (Ultimately, Omar Khayyam developed a general method to find the real roots of any cubic polynomial.) However, they seem to have implicitly believed that all real numbers were expressible using such roots (radical number?s), which we now know is false even for some algebraic numbers, such as the real root of . In any case, they only used such numbers.
Later, European mathematicians of the early modern era (particularly Cardano?, Tartaglia?, and Ferrari?) had begun work with imaginary numbers, which are necessarily irrational. Following this, Lambert? and Legendre succeeded in proving the irrationality of pi, e, and their powers, which ultimately led to the conjecture that they were transcendental (whereas radical numbers, and even the root of , are by definition algebraic); this conjecture was later established by Hermite and Lindemann?. Around this time, Euler and Lagrange popularized continued fractions (see below) to study both rational and irrational numbers.
During the arithmetization of analysis? in the 19th century, people sometimes wrote of the problem of ‘defining irrational numbers’. The actual issue here was defining real numbers in general; one could define rational numbers algebraically, leaving only the irrational numbers as the problem. However, this may be a red herring; one could just as easily define algebraic numbers algebraically and say that the problem is defining transcendental numbers; indeed, it was only with the discovery that such numbers as and are irrational that work on this problem came to life. On the other hand, it's not clear that anybody could completely work out the order properties of algebraic numbers without already coming upon Dedekind's solution. In any case, specific irrational algebraic numbers such as posed no difficulty to the finitist methods used by such algebraists as Leopold Kronecker.
To this day, there are various specific real numbers (such as , the Euler–Mascheroni constant , etc.) whose rationality or irrationality is unknown. In constructive mathematics, this makes it unproved that these numbers are rational or irrational (although the double negation of this statement can be proved for any real number). The question of whether is rational or irrational is part of a famous illustration of the nature of constructive vs nonconstructive proof. (Namely, there is a cheap and easy nonconstructive proof that there exist irrational numbers and such that is rational: let be and let be either or , depending on whether the latter is rational or irrational. A constructive proof that decides which of these is the case is much harder: turns out to be irrational, by a constructive version of the Gelfond–Schneider theorem?.1)
The Baire space is homeomorphic to the product space of copies of the discrete space of natural numbers. The homeomorphism is given by continued fractions (see below).
Every inhabited Polish space is a quotient space of , and is itself a Polish space.
As a subset of the real line, is a full set (meaning that its complement, the set of rational numbers, is null).
Cantor space may be identified with a subspace of , consisting of those irrational numbers whose continued fraction expansion consists only of and (but this does not agree with the usual inclusions into ).
The fan theorem states precisely that (when thought of as a topological space) is sober or that (when thought of as a locale) is topological/spatial/has enough points. This is true in classical mathematics and in intuitionistic mathematics but fails in other forms of constructive mathematics.
(Main article: continued fraction.)
Let be an infinite sequence of integers, all positive except (possibly) . We interpret this as the number
By truncating this expression after , we produce a rational number; altogether, this is an infinite sequence of rational numbers.
This is a Cauchy sequence whose limit is irrational. Furthermore, every irrational number has a unique representation in this way. Yet more, the bijection thus shown between and the infinitary cartesian product is a homeomorphism when the two sets are given their usual topologies.
The usual proofs of these theorems are entirely constructive. Accordingly, in the foundations of mathematics, one may define Baire space either as the space of irrational numbers or as the infinite product . However, to treat Baire space as a uniform space or as a metric space, one uses the structure from .
For the definition of the irrational numbers in terms of non-repeating base 10 infinite radix expansions
Nichols, Eugene D, et al. Holt Algebra with Trigonometry. Holt, Rinehart and Winston : Harcourt Brace Jovanovich, 1992.
Marecek, Lynn, et al. Prealgebra 2e. OpenStax, Rice University, 2020.
That Cauchy irrational numbers have radix expansions in constructive mathematics
Of course, one could also take and , which are both irrational by easy constructive proofs, if one is after definite irrational numbers and such that is rational (here the result is ). The irrationality of follows, as easily as that of , from the fundamental theorem of arithmetic: there do not exist nonzero integers such that . ↩
Last revised on August 4, 2024 at 20:19:33. See the history of this page for a list of all contributions to it.