For numbers "constructible" in the sense of set theory, see Constructible universe.
The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1 and is therefore a constructible number
In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only integers and the operations for addition, subtraction, multiplication, division, and square roots.
The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. A point is constructible if it can be produced as one of the points of a compass and straight edge construction (an endpoint of a line segment or crossing point of two lines or circles), starting from a given unit length segment. Alternatively and equivalently, taking the two endpoints of the given segment to be the points (0, 0) and (1, 0) of a Cartesian coordinate system, a point is constructible if and only if its Cartesian coordinates are both constructible numbers.[1] Constructible numbers and points have also been called ruler and compass numbers and ruler and compass points, to distinguish them from numbers and points that may be constructed using other processes.[2]
The set of constructible numbers forms a field: applying any of the four basic arithmetic operations to members of this set produces another constructible number. This field is a field extension of the rational numbers and in turn is contained in the field of algebraic numbers.[3] It is the Euclidean closure of the rational numbers, the smallest field extension of the rationals that includes the square roots of all of its positive numbers.[4]
The proof of the equivalence between the algebraic and geometric definitions of constructible numbers has the effect of transforming geometric questions about compass and straightedge constructions into algebra, including several famous problems from ancient Greek mathematics. The algebraic formulation of these questions led to proofs that their solutions are not constructible, after the geometric formulation of the same problems previously defied centuries of attack.
^Kazarinoff (2003, pp. 10 & 15); Martin (1998), Corollary 2.16, p. 41.
^Martin (1998), pp. 31–32.
^Courant & Robbins (1996), Section III.2.2, "All constructible numbers are algebraic", pp. 133–134.
^Kazarinoff (2003), p. 46.
and 24 Related for: Constructible number information
coordinate system, a point is constructible if and only if its Cartesian coordinates are both constructible numbers. Constructible numbers and points have also...
infinitely many constructible polygons, but only 31 with an odd number of sides are known. Some regular polygons are easy to construct with compass and...
positive rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related...
that can be constructed with compass and straightedge Constructiblenumber, a complex number associated to a constructible point Constructible polygon, a...
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L {\displaystyle L} , is a particular class of...
are the two roots of the quadratic x2 − 2ax + a2 + b2. A constructiblenumber can be constructed from a given unit length using a straightedge and compass...
Equivalently, a regular n-gon is constructible if and only if the cosine of its common angle is a constructiblenumber—that is, can be written in terms...
Fermat prime, the regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass. It is the...
a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructiblenumber: A number representing...
{\sqrt[{3}]{2}}} is not a constructiblenumber. This is a consequence of the fact that the coordinates of a new point constructed by a compass and straightedge...
There are angles that are not constructible but are trisectible (despite the one-third angle itself being non-constructible). For example, 3π/7 is such...
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written...
^{-1}.} As the root of a quadratic polynomial, the golden ratio is a constructiblenumber. The conjugate root to the minimal polynomial x 2 − x − 1 {\displaystyle...
numbers of sufficient precision, such as the iRRAM package. Constructiblenumber Definable number Semicomputable function Transcomputational problem van der...
concept of space-constructible functions. The deterministic and nondeterministic space hierarchy theorems state that for all space-constructible functions f(n)...
that the higher direct images of a constructible sheaf are constructible. Here we use the definition of constructible étale sheaves from the book by Freitag...
Wantzel proved it to be impossible because the cube root of 2 is not a constructiblenumber. The cube has three uniform colorings, named by the unique colors...
straightedge and compass, the constructible numbers are those complex numbers whose real and imaginary parts can be constructed using straightedge and compass...
infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously. There...
way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers. The trigonometric functions...
construction of square roots (see constructiblenumber), since starting with a rectangle that has a width of 1 the constructed square will have a side length...
constructions (see constructiblenumber). In particular it is important to assure that for two given line segments, a new line segment can be constructed, such that...
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