Part II, Chapter I — Cantor’s Demonstration of the Existence of Transcendental Numbers¶
Part II: Transcendental Numbers and the Quadrature of the Circle — Chapter I (§§ 1–3; Ginn & Co., 1897, pp. 49–54)
§ 1. Rational, Algebraic, and Transcendental Numbers on the Continuum¶
1. Represent real numbers as usual by points along the axis of abscissas. If we restrict ourselves to rational numbers, the corresponding points fill the axis densely throughout (überall dicht)—in every interval, however small, infinitely many rational points lie. Nevertheless, as the ancients already discovered, the continuum of points on the line is not exhausted by rationals: between them enter the irrational numbers, and we must ask whether fundamental distinctions exist within the irrationals themselves.
We first define algebraic numbers: every real root $\omega$ of an algebraic equation with integer coefficients,
is called an algebraic number (rational numbers arising as the special case $a_0 \omega + a_1 = 0$ of degree $n = 1$). Now we pose the decisive question: Does the totality of all real algebraic numbers exhaust the real continuum, or does it still form only a discrete, perforated set into whose gaps new numbers may be inserted? Such non-algebraic real numbers—incapable of satisfying any polynomial equation with integer coefficients—are called transcendental numbers.
This question was first answered affirmatively by Joseph Liouville (Comptes rendus, 1844; Journal de Liouville, Vol. XVI, 1851), who proved the existence of transcendental numbers by means of continued fractions. Liouville’s method, however, is rather intricate; the investigation becomes remarkably simple through the set-theoretic principles created by Georg Cantor in his foundational memoir Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen (Crelle's Journal, Vol. 77, 1874 [1873]), combined with the diagonal simplification Cantor presented before the Naturforscherversammlung at Halle in 1891.
§ 2. Countability, Power (Mächtigkeit), and Algebraic Height¶
2. Cantor's proof rests on showing that the real algebraic numbers form a countable (*abzählbare*) aggregate, whereas the continuum does not. Countability means that all real algebraic numbers can be arranged in a definite numbered sequence $\omega_1, \omega_2, \omega_3, \dots$ in one-to-one correspondence with the positive integers $1, 2, 3, \dots$.
At first sight a paradox seems to arise: the positive integers form only a proper part of the algebraic numbers, yet pairing them one-to-one seems to make the part equal to the whole! That objection rests on a false analogy with finite sets: for infinite aggregates, the Euclidean maxim that the part is smaller than the whole loses its meaning—just as the positive integers $1, 2, 3, \dots, n, \dots$ stand in exact one-to-one correspondence with the positive even integers $2, 4, 6, \dots, 2n, \dots$. In place of “larger” and “smaller,” Cantor introduces power (Mächtigkeit): two infinite aggregates have the same power if they can be placed in one-to-one correspondence.
To prove that the totality of real algebraic numbers has the same power as $\mathbb{Z}^+$, write every algebraic number as a real root of its unique irreducible integer equation
and define the algebraic height $N$ of the equation by
For each positive integer $N$, only finitely many equations—and hence only finitely many real algebraic numbers, say $\varphi(N)$ in number—possess height $N$, since $n \le N$ and $N - (n - 1)$ admits only finitely many decompositions into non-negative integers $|a_i|$. Discarding reducible equations and complex roots yields the opening rows of Cantor’s enumeration:
$N$ |
$n$ |
$(|a_0|, |a_1|, \dots)$ |
Irreducible Equations (Real Roots) |
$\varphi(N)$ |
Real Algebraic Roots $\omega$ |
|---|---|---|---|---|---|
$1$ |
$1$ |
$(1, 0)$ |
$\omega = 0$ |
$1$ |
$0$ |
$2$ |
$1$ |
$(1, 1)$ |
$\omega \pm 1 = 0$ |
$2$ |
$-1,\; +1$ |
$3$ |
$1$ |
$(1, 2),\; (2, 1)$ |
$\omega \pm 2 = 0,\; 2\omega \pm 1 = 0$ (note $\omega^2 + 1 = 0$ has no real roots) |
$4$ |
$-2,\; -\frac{1}{2},\; +\frac{1}{2},\; +2$ |
$4$ |
$1, 2$ |
$(1, 3), (3, 1)$; $(1, 1, 1), (1, 0, 2), (2, 0, 1)$ |
$\omega \pm 3 = 0$, $3\omega \pm 1 = 0$, $\omega^2 \pm \omega - 1 = 0$, $\omega^2 - 2 = 0$, $2\omega^2 - 1 = 0$ |
$12$ |
$\pm 3,\; \pm\frac{1}{3},\; \frac{\pm 1 \pm \sqrt{5}}{2},\; \pm\sqrt{2},\; \pm\frac{1}{\sqrt{2}}$ |
Ordering roots within each finite height class $N = 1, 2, 3, 4, \dots$ by increasing magnitude enumerates every real algebraic number in a single sequence $\omega_1, \omega_2, \omega_3, \dots$.
§ 3. Cantor’s Diagonal Construction of Transcendental Numbers¶
3. Conversely, the continuum of all real numbers—even within an arbitrarily tiny interval—is not countable, and Cantor’s 1891 diagonal procedure immediately exhibits infinitely many transcendental numbers inside any prescribed neighborhood.
Write every real algebraic number $\omega_1, \omega_2, \omega_3, \dots$ in infinite decimal form (noting that any decimal not ending in an infinite string of $9$’s has a unique representation). Suppose we wish to construct transcendental numbers sharing a prescribed $5$-decimal prefix $0.c_1 c_2 c_3 c_4 c_5\dots$:
Choose the $6$-th decimal digit different from $9$ and different from the $6$-th decimal digit of $\omega_1$;
Choose the $7$-th decimal digit different from $9$ and different from the $7$-th decimal digit of $\omega_2$;
In general, choose the $(5 + k)$-th decimal digit different from $9$ and different from the $(5 + k)$-th decimal digit of $\omega_k$.
The resulting decimal fraction never terminates in repeating $9$’s and differs from every algebraic number $\omega_k$ in our complete table at the $(5 + k)$-th decimal place! Hence it is necessarily a transcendental number. Indeed, because at every decimal step (avoiding $9$ and one diagonal digit) we retain a free choice among eight digits, we can form—so to speak—$8^\infty$ transcendental numbers inside any interval, however small: there are infinitely more transcendental numbers than algebraic numbers.