Irrational numbers

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 collection of all irrational real numbers is interesting in topology and constructive analysis.

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

Definition

An irrational (real) number is a real number $x$ such that, given any rational number $a$ (thought of as a real number), the absolute value ${|x - a|}$ is positive. (This precise definition is used in constructive mathematics; classically this is equivalent to saying that $x \ne a$.)

The set of irrational 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.)

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 real numbers) attributed to Eudoxus? in Book X of Euclid's Elements?.

The mediaeval Arabic mathematicians were the first to treat irrational numbers algebraically as numbers (rather than geometrically as ratios of magnitudes); they applied algebra to square roots, cube roots, etc. However, they seem to have implicitly believed that all real numbers were expressible using such roots, which we now know is false even for some algebraic numbers, such as the root of $x^5 + 2x + 1$. In any case, they only used such numbers.

In the early modern era, Latin mathematicians began work with imaginary number?s, which are necessarily irrational. They subsequently proved the irrationality of pi?, e?, and their powers, which ultimately led to the discovery that they were transcendental (whereas square roots, and even the root of $x^5 + 2x + 1$, are by definition algebraic). Leonhard Euler and Joseph Lagrange? popularised continued fraction?s (see below) to study both rational and irrational numbers.

During 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 Richard Dedekind’s solution. In any case, specific irrational algebraic numbers such as $\sqrt{2}$ posed no difficulty to 19th-century constructive algebraists such as Leopold Kronecker?.

To this day, there are various specific real numbers (such as $\pi + \mathrm{e}$, the Euler 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. (There is an 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 is much harder, but in the end it is more specific; once we prove that ${\sqrt{2}}^{\sqrt{2}}$ is irrational, say with a constructive version of the Gelfond-Schneider theorem?, we know what $a$ should be.)

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 fraction?s (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

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 $\mathbb{N}^{\mathbb{N}}$.

References

• Wikipedia (English):

Revised on June 27, 2011 12:29:29 by Toby Bartels (76.85.192.183)