nLab irrational number

Idea

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:

• Historically, it was an important discovery that irrational real numbers exist.
• The topological space of all irrational real numbers is interesting in constructive analysis, computability theory, and descriptive set theory.

Of course, there are also various theorems about general classes of numbers that distinguish rational from irrational numbers.

Definition

In the real numbers

An affine function on the real numbers consists of a function $f$ and real number coefficients $a$ and $b$ such that for all real numbers $x$, $f(x) = a x + b$. The constant function at zero $\lambda x.0$ is an affine function where $a = 0$ and $b = 0$. A real number is irrational if for all affine functions $f$ with integer coefficients, if $f(x) = 0$, then $f = \lambda x.0$. This is equivalent to saying that a real number is irrational if for all rational numbers $a$, $x \neq a$.

Alternatively, a real number $x$ is irrational if given any rational number $a$ (thought of as a real number), the absolute value ${|x - a|}$ 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 $\mathbb{I}$, $\mathbb{J}$, or $\mathbb{B}$ (in various fonts). The $\mathbb{I}$ and $\mathbb{J}$ stand for ‘irrational’, while the $\mathbb{B}$ stands for ‘Baire’ (see the next paragraph). Here we will use $\mathbb{J}$.

We may give $\mathbb{J}$ a topology as a subspace of the real line $\mathbb{R}$. With this topology, $\mathbb{J}$ 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, $\mathbb{J}$ is an example of such a space.)

As non-repeating radix expansions

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 $b \gt 1$ denote the base of the radix expansion, $[0, b)$ denote the half-open interval in the natural numbers of all natural numbers less than $b$. Base $b$ infinite radix expansions are elements of $\mathbb{Z} \times [0, b)^\mathbb{N}$, with the idea that each pair $(i, d)$ consists of an integer $i$ and a sequence of digits $d(n)$ in the base $b$ infinite radix expansion. The series

$$\sum_{n = 0}^\infty i + \frac{d(n)}{b^{n + 1}}$$

can be shown to be a Cauchy sequence.

The set of repeating base $b$ infinite radix expansion is the subset of $\mathbb{Z} \times [0, b)^\mathbb{N}$ such that for pairs $(i, d)$ in the subset, there exist natural number $k$ and positive natural number $n$ such that the sequence $\lambda m.d(m + k)$ factors through the cyclic group $\mathbb{Z}/n\mathbb{Z}$.

$$\lambda m.d(m + k):\mathbb{N} \to \mathbb{Z}/n\mathbb{Z} \to [0, b)$$

Two base $b$ infinite radix expansions $(i, d)$ and $(j, e)$ are said to be apart from each other if $i \neq j$ or there exists a natural number $n$ such that $d(n) \neq e(n)$

$$(i, d) \# (j, e) \coloneqq (i \neq j) \vee \exists n \in \mathbb{N}.d(n) \neq e(n)$$

A base $b$ infinite radix expansion $x$ is strictly non-repeating if it is apart from every repeating base $b$ infinite radix expansion. An irrational number is a strictly non-repeating base $b$ infinite radix expansion.

In constructive mathematics, not every strictly irrational number is the limit of the series

$$\sum_{n = 0}^\infty i + \frac{d(n)}{b^{n + 1}}$$

given by a non-repeating base $b$ 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 $b$ infinite radix expansions.

In Archimedean integral domains

Let $R$ be an Archimedean integral domain with the integers $\mathbb{Z} \subseteq R$ being a integral subdomain of $R$.

An element $r \in R$ is irrational if for all $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, $\vert a \vert \gt 0$ and $\vert a \cdot r - b \vert \gt 0$.

The set of irrational numbers in $R$ is defined as

$$\mathbb{J}_R \coloneqq \{r \in R \vert \forall a, b \in \mathbb{Z}. (\vert a \vert \gt 0) \wedge (\vert a \cdot r - b \vert \gt 0) \}$$

In integral domains with a p-adic norm

Let $R$ be an integral domain with a p-adic norm $\vert(-)\vert_p$ for a prime number $p$, with the integers $\mathbb{Z} \subseteq R$ being a integral subdomain of $R$.

An element $r \in R$ is irrational if for all $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, $\vert a \vert_p \gt 0$ and $\vert a \cdot r - b \vert_p \gt 0$.

The set of irrational numbers in $R$ is defined as

$$\mathbb{J}_R \coloneqq \{r \in R \vert \forall a, b \in \mathbb{Z}. (\vert a \vert_p \gt 0) \wedge (\vert a \cdot r - b \vert_p \gt 0) \}$$

History

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) $x$ and $y$ are commensurable? in the sense that there exists a unit length $u$ such that $x = m u$ and $y = n u$ for some natural numbers $m$ and $n$. 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 $x^5 + 2 x + 1$. 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 $x^5 + 2x + 1$, 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 $\pi$ and $\mathrm{e}$ 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 $\sqrt{2}$ 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 $\pi + \mathrm{e}$, the Euler–Mascheroni constant $\gamma$, 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 ${\sqrt{2}}^{\sqrt{2}}$ 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 $a$ and $b$ such that $a^b$ is rational: let $b$ be $\sqrt{2}$ and let $a$ be either $\sqrt{2}$ or ${\sqrt{2}}^{\sqrt{2}}$, depending on whether the latter is rational or irrational. A constructive proof that decides which of these is the case is much harder: ${\sqrt{2}}^{\sqrt{2}}$ turns out to be irrational, by a constructive version of the Gelfond–Schneider theorem?.¹)

Properties

The Baire space $\mathbb{J}$ is homeomorphic to the product space $\mathbb{N}^{\mathbb{N}}$ of $\aleph_0$ 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 $\mathbb{J}$, and $\mathbb{J}$ is itself a Polish space.

As a subset of the real line, $\mathbb{J}$ is a full set (meaning that its complement, the set of rational numbers, is null).

Cantor space may be identified with a subspace of $\mathbb{J}$, consisting of those irrational numbers whose continued fraction expansion consists only of $1$ and $2$ (but this does not agree with the usual inclusions into $\mathbb{R}$).

The fan theorem states precisely that $\mathbb{J}$ (when thought of as a topological space) is sober or that $\mathbb{J}$ (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.

Continued fractions

(Main article: continued fraction.)

Let $[a_0;a_1,a_2,a_3,\ldots]$ be an infinite sequence of integers, all positive except (possibly) $a_0$. We interpret this as the number

$$a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}} .$$

By truncating this expression after $a_i$, we produce a rational number; altogether, this is an infinite sequence of rational numbers.

Theorems

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 $\mathbb{J}$ and the infinitary cartesian product $\mathbb{Z} \times \mathbb{N}^+ \times \mathbb{N}^+ \times \mathbb{N}^+ \times \cdots$ 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 $\mathbb{N}^{\mathbb{N}}$. However, to treat Baire space as a uniform space or as a metric space, one uses the structure from $\mathbb{N}^{\mathbb{N}}$.

Examples

• pi

• e

• golden ratio

References

• Wikipedia (English):
  • Baire space (set theory)
  • Irrational numbers → History

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

• Andrew Swan (2024) on Category Theory Zulip, Radix expansions in constructive mathematics

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1. Of course, one could also take $a = \sqrt{2}$ and $b = 2\frac{\log 3}{\log 2})$, which are both irrational by easy constructive proofs, if one is after definite irrational numbers $a$ and $b$ such that $a^b$ is rational (here the result is $a^b = 3$). The irrationality of $\frac{\log 3}{\log 2}$ follows, as easily as that of $\sqrt{2}$, from the fundamental theorem of arithmetic: there do not exist nonzero integers $p, q$ such that $2^p = 3^q$. ↩