Constructible number

If ${\displaystyle a}$ and ${\displaystyle b}$ are the non-zero lengths of constructed segments then elementary compass and straightedge constructions can be used to obtain constructed segments of lengths ${\displaystyle a+b}$, ${\displaystyle |a-b|}$, ${\displaystyle ab}$, and ${\displaystyle a/b}$. The latter two can be done with a construction based on the intercept theorem. A slightly less elementary construction using these tools is based on the geometric mean theorem and will construct a segment of length ${\displaystyle {\sqrt {a}}}$ from a constructed segment of length ${\displaystyle a}$.[11]

Compass and straightedge constructions for constructible numbers

${\displaystyle ab}$ based on the intercept theorem

${\displaystyle {\frac {a}{b}}}$ based on the intercept theorem

${\displaystyle {\sqrt {p}}}$ based on the geometric mean theorem

It follows from these constructions that every algebraically constructible number is geometrically constructible.

In the other direction, a set of geometric objects may be specified by algebraically constructible real numbers: coordinates for points, slope and ${\displaystyle y}$-intercept for lines, and center and radius for circles. It is possible (but tedious) to develop formulas in terms of these values, using only arithmetic and square roots, for each additional object that might be added in a single step of a compass-and-straightedge construction. It follows from these formulas that every geometrically constructible number is algebraically constructible.[12]

The definition of algebraically constructible numbers includes the sum, difference, product, and multiplicative inverse of any of these numbers, the same operations that define a field in abstract algebra. Thus, the constructive numbers (defined in any of the above ways) form a field. More specifically, the constructive real numbers form a Euclidean field, an ordered field containing a square root of each of its positive elements.[13] Examining the properties of this field and its subfields leads to necessary conditions on a number to be constructible, that can be used to show that specific numbers arising in classical geometric construction problems are not constructible.

It is convenient to consider, in place of the whole field of constructible numbers, the subfield ${\displaystyle \mathbb {Q} (\gamma )}$ generated by any given constructible number ${\displaystyle \gamma }$, and to use the algebraic construction of ${\displaystyle \gamma }$ to decompose this field. If ${\displaystyle \gamma }$ is a constructible real number, then the values occurring within a formula constructing it can be used to produce a finite sequence of real numbers ${\displaystyle \alpha _{1},\dots ,a_{n}=\gamma }$ such that, for each ${\displaystyle i}$, ${\displaystyle \mathbb {Q} (\alpha _{1},\dots ,a_{i})}$ is an extension of ${\displaystyle \mathbb {Q} (\alpha _{1},\dots ,a_{i-1})}$ of degree 2.[14] Using slightly different terminology, a real number is constructible if and only if it lies in a field at the top of a finite tower of real quadratic extensions,

${\displaystyle \mathbb {Q} =K_{0}\subseteq K_{1}\subseteq \dots \subseteq K_{n},}$

starting with the rational field ${\displaystyle \mathbb {Q} }$ where ${\displaystyle \gamma }$ is in ${\displaystyle K_{n}}$ and for all ${\displaystyle 0<j\leq n}$, ${\displaystyle [K_{j}:K_{j-1}]=2}$.[15] It follows from this decomposition that the degree of the field extension ${\displaystyle [\mathbb {Q} (\gamma ):\mathbb {Q} ]}$ is ${\displaystyle 2^{r}}$, where ${\displaystyle r}$ counts the number of quadratic extension steps.

Analogously to the real case, a complex number is constructible if and only if it lies in a field at the top of a finite tower of complex quadratic extensions.[16] More precisely, ${\displaystyle \gamma }$ is constructible if and only if there exists a tower of fields

${\displaystyle \mathbb {Q} =F_{0}\subseteq F_{1}\subseteq \dots \subseteq F_{n},}$

where ${\displaystyle \gamma }$ is in ${\displaystyle F_{n}}$, and for all ${\displaystyle 0<j\leq n}$, ${\displaystyle [F_{j}:F_{j-1}]=2}$. The difference between this characterization and that of the real quadratic numbers is only that the fields in this tower are not restricted to being real. Consequently, if a complex number ${\displaystyle \gamma }$ is constructible, then ${\displaystyle [\mathbb {Q} (\gamma ):\mathbb {Q} ]}$ is a power of two. However, this necessary condition is not sufficient: there exist field extensions whose degree is a power of two that cannot be factored into a sequence of quadratic extensions.[17]

The fields that can be generated in this way from towers of quadratic extensions of ${\displaystyle \mathbb {Q} }$ are called iterated quadratic extensions of ${\displaystyle \mathbb {Q} }$. The fields of real and complex constructible numbers are the unions of all real or complex iterated quadratic extensions of ${\displaystyle \mathbb {Q} }$.[18]

Trigonometric numbers are irrational cosines or sines of angles that are rational multiples of ${\displaystyle \pi }$. Such a number is constructible if and only if the denominator of the fully reduced multiple is a power of 2 or the product of a power of 2 with the product of one or more distinct Fermat primes. Thus, for example, ${\displaystyle \cos(\pi /15)}$ is constructible because 15 is the product of two Fermat primes, 3 and 5.

For a list of trigonometric numbers expressed in terms of square roots see trigonometric constants expressed in real radicals.

Although duplication of the cube is impossible, duplication of the square is not.

The ancient Greeks thought that certain problems of straightedge and compass construction they could not solve were simply obstinate, not unsolvable.[19] However, the non-constructibility of certain numbers proves them to be logically impossible to perform. (The problems themselves, however, are solvable using methods that go beyond the constraint of working only with straightedge and compass, and the Greeks knew how to solve them in this way.)

In the following chart, each table represents a specific ancient construction problem. The left column gives the name of the problem. The second column gives an equivalent algebraic formulation of the problem. In other words, the solution to the problem is affirmative if and only if each number in the given set of numbers is constructible. Finally, the last column provides a simple counterexample. In other words, the number in the last column is an element of the set in the same row, but is not constructible.

Construction problem	Associated set of numbers	Counterexample
Doubling the cube	${\displaystyle \left\{{\sqrt[{3}]{x}}:x{\text{ is constructible}}\right\}}$	${\displaystyle {\sqrt[{3}]{2}}}$ (the edge length of a doubled unit cube) is not constructible, because its minimal polynomial has degree 3 over Q[20]
Trisecting the angle	${\displaystyle \left\{\cos \left({\frac {\arccos x}{3}}\right):x{\text{ is constructible}}\right\}}$	${\displaystyle \cos \left({\frac {\arccos(1/2)}{3}}\right)=\cos 20^{o}}$ (one of the coordinates of a unit-length segment trisecting an axis-aligned equilateral triangle) is not constructible, because ${\displaystyle \cos 20^{o}}$ has minimal polynomial of degree 3 over Q[20]
Squaring the circle	${\displaystyle \left\{r{\sqrt {\pi }}:r{\text{ is constructible}}\right\}}$	${\displaystyle {\sqrt {\pi }}}$ (the side length of a square with the same area as a unit circle) is not constructible, because it is not algebraic over Q[20]
Constructing regular polygons	${\displaystyle \left\{\cos 2\pi /n:n\in \mathbb {N} ,n\geq 3\right\}}$	${\displaystyle \cos 2\pi /7}$ (the x-coordinate of a vertex of an axis-aligned regular heptagon) is not constructible, because 7 is not a Fermat prime, nor is 7 the product of ${\displaystyle 2^{k}}$ and one or more distinct Fermat primes[21]

The birth of the concept of constructible numbers is inextricably linked with the history of the three impossible compass and straightedge constructions: duplicating the cube, trisecting an angle, and squaring the circle. The restriction of using only compass and straightedge in geometric constructions is often credited to Plato due to a passage in Plutarch. According to Plutarch, Plato gave the duplication of the cube (Delian) problem to Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry (Plut., Quaestiones convivales VIII.ii, 718ef). However, this attribution is challenged,[22] due, in part, to the existence of another version of the story (attributed to Eratosthenes by Eutocius of Ascalon) that says that all three found solutions but they were too abstract to be of practical value.[23] Since Oenopides (circa 450 BCE) is credited with two ruler and compass constructions, by Proclus– citing Eudemus (circa 370 - 300 BCE)–when other methods were available to him, has led some authors to hypothesize that Oenopides originated the restriction.

The restriction to compass and straightedge is essential in making these constructions impossible. Angle trisection, for instance, can be done in many ways, several known to the ancient Greeks. The Quadratrix of Hippias of Elis, the conics of Menaechmus, or the marked straightedge (neusis) construction of Archimedes have all been used, as has a more modern approach via paper folding.

Although not one of the classic three construction problems, the problem of constructing regular polygons with straightedge and compass is usually treated alongside them. The Greeks knew how to construct regular n-gons with n = 2^h, 3, 5 (for any integer h ≥ 2) or the product of any two or three of these numbers, but other regular n-gons eluded them. Then, in 1796, an eighteen-year-old student named Carl Friedrich Gauss announced in a newspaper that he had constructed a regular 17-gon with straightedge and compass.[24] Gauss's treatment was algebraic rather than geometric; in fact, he did not actually construct the polygon, but rather showed that the cosine of a central angle was a constructible number. The argument was generalized in his 1801 book Disquisitiones Arithmeticae giving the sufficient condition for the construction of a regular n-gon. Gauss claimed, but did not prove, that the condition was also necessary and several authors, notably Felix Klein,[25] attributed this part of the proof to him as well.[26]

Pierre Wantzel (1837) proved algebraically that the problems of doubling the cube and trisecting the angle are impossible to solve if one uses only compass and straightedge. In the same paper he also solved the problem of determining which regular polygons are constructible: a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e., the sufficient conditions given by Gauss are also necessary)

An attempted proof of the impossibility of squaring the circle was given by James Gregory in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of π. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that π is a transcendental number.

The study of constructible numbers, per se, was initiated by René Descartes in La Géométrie, an appendix to his book Discourse on the Method published in 1637. Descartes associated numbers to geometrical line segments in order to display the power of his philosophical method by solving an ancient straightedge and compass construction problem put forth by Pappus.[27]

• Computable number
• Definable real number

• Boyer, Carl B. (2004) [1956], History of Analytic Geometry, Dover, ISBN 978-0-486-43832-0
• Fraleigh, John B. (1994), A First Course in Abstract Algebra (5th ed.), Addison Wesley, ISBN 978-0-201-53467-2
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• Kazarinoff, Nicholas D. (2003) [1970], Ruler and the Round: Classic Problems in Geometric Constructions, Dover, ISBN 0-486-42515-0
• Klein, Felix (1956) [1930], Famous Problems of Elementary Geometry, Dover
• Knorr, Wilbur Richard (1986), The Ancient Tradition of Geometric Problems, Dover Books on Mathematics, Courier Dover Publications, ISBN 9780486675329
• Martin, George E. (1998), Geometric Constructions, Undergraduate Texts in Mathematics, Springer-Verlag, New York, doi:10.1007/978-1-4612-0629-3, ISBN 0-387-98276-0, MR 1483895
• Moise, Edwin E. (1974), Elementary Geometry from an Advanced Standpoint (2nd ed.), Addison Wesley, ISBN 0-201-04793-4
• Roman, Steven (1995), Field Theory, Springer-Verlag, ISBN 978-0-387-94408-1
• Rotman, Joseph J. (2006), A First Course in Abstract Algebra with Applications (3rd ed.), Prentice Hall, ISBN 978-0-13-186267-8
• Stewart, Ian (1989), Galois Theory (2nd ed.), Chapman and Hall, ISBN 978-0-412-34550-0
• Wantzel, P. L. (1837), "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas", Journal de Mathématiques Pures et Appliquées, 1 (2): 366–372

Wikimedia Commons has media related to Constructible numbers.

• Chris Cooper: Galois Theory. Lectures notes, Macquarie University, §6 Ruler and Compass Constructability, pp. 55-63
• Weisstein, Eric W., "Constructible Number", MathWorld
• Constructible Numbers at Cut-the-knot