In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Cramer's paradox or the Cramer–Euler paradox
[Weisstein, Eric W. "Cramér-Euler Paradox." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cramer-EulerParadox.html] is the statement that the number of points of intersection of two higher-order curves in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
can be greater than the number of arbitrary points that are usually needed to define one such curve. It is named after the
Genevan
, neighboring_municipalities= Carouge, Chêne-Bougeries, Cologny, Lancy, Grand-Saconnex, Pregny-Chambésy, Vernier, Veyrier
, website = https://www.geneve.ch/
Geneva ( ; french: Genève ) frp, Genèva ; german: link=no, Genf ; it, Ginevr ...
mathematician
Gabriel Cramer
Gabriel Cramer (; 31 July 1704 – 4 January 1752) was a Genevan mathematician. He was the son of physician Jean Cramer and Anne Mallet Cramer.
Biography
Cramer showed promise in mathematics from an early age. At 18 he received his doctorate ...
.
This phenomenon appears paradoxical because the points of intersection fail to uniquely define any curve (they belong to at least two different curves) despite their large number.
It is the result of a naive understanding or a misapplication of two theorems:
*
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the degr ...
states that the number of points of intersection of two
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s is equal to the product of their degrees, provided that certain necessary conditions are met. In particular, two curves of degree
generally have
points of intersection.
*
Cramer's theorem states that a curve of degree
is determined by
points, again assuming that certain conditions hold.
For all
,
, so it would naively appear that for degree three or higher, the intersection of two curves would have enough points to define either of the curves uniquely. However, because these points belong to both curves, they do not define a unique curve of this degree. The resolution of the paradox is that the
bound on the number of points needed to define a curve only applies to points in
general position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that ar ...
. In certain
degenerate
Degeneracy, degenerate, or degeneration may refer to:
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* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
cases,
points are not enough to determine a curve uniquely.
History
The paradox was first published by
Colin Maclaurin
Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bei ...
. Cramer and
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer.
It has become known as ''Cramer's paradox'' after featuring in his 1750 book ''Introduction à l'analyse des lignes courbes algébriques'', although Cramer quoted Maclaurin as the source of the statement. At about the same time, Euler published examples showing a cubic curve which was not uniquely defined by 9 points
and discussed the problem in his book ''
Introductio in analysin infinitorum
''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...
''. The result was publicized by
James Stirling and explained by
Julius Plücker
Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the disc ...
.
[
]
No paradox for lines and nondegenerate conics
For first order curves (that is lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
) the paradox does not occur, because , so