In mathematics, Cramer's paradox is the statement that the number of points of intersection of two higher-order curves can be greater than the number of arbitrary points that are usually needed to define one such curve. It is named after the Swiss mathematician Gabriel Cramer.
This paradox is the result of a naive understanding or a misapplication of two theorems:
- Bézout's theorem (the number of points of intersection of two algebraic curves is equal to the product of their degrees, provided that certain necessary conditions are met).
- Cramer's theorem (a curve of degree n is determined by n(n + 3)/2 points, again assuming that certain conditions hold).
Observe that ∀n ≥ 3, n2 ≥ n(n + 3)/2.
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.
No paradox for lines and nondegenerate conics
For first order curves (that is lines) the paradox does not occur. In general two lines L1 and L2 intersect at a single point P unless the lines are of equal gradient. A single point is not sufficient to define a line (two are needed); through the point P there pass not only the two given lines but an infinite number of other lines as well.
Similarly two nondegenerate conics intersect at most at 4 points, and 5 points are needed to define a nondegenerate conic.
The Cramer's example for cubic curves
In a letter to Euler, Cramer pointed out that the cubic curves y3-y=0 and x3-x=0 intersect in precisely 9 points. Hence 9 points are not sufficient to uniquely determine a cubic curve.
- Maclaurin, Colin (1720). Geometria Organica. London.
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- Tweedie, Charles (1915). "A Study of the Life and Writings of Colin Maclaurin". The Mathematical Gazette 8 (119): 133–151. JSTOR 3604693.
- Euler, L. "Sur une contradiction apparente dans la doctrine des lignes courbes." Mémoires de l'Academie des Sciences de Berlin 4, 219-233, 1750
- Weisstein, Eric W., "Cramér-Euler Paradox", MathWorld.