Elliptic Curve - Elliptic Curves Over The Rational Numbers

Elliptic Curves Over The Rational Numbers

A curve E defined over the field of rational numbers is also defined over the field of real numbers, therefore the law of addition (of points with real coordinates) by the tangent and secant method can be applied to E. The explicit formulae show that the sum of two points P and Q with rational coordinates has again rational coordinates, since the line joining P and Q has rational coefficients. This way, one shows that the set of rational points of E forms a subgroup of the group of real points of E. As this group, it is an abelian group, that is, P + Q = Q + P.

Read more about this topic:  Elliptic Curve

Famous quotes containing the words curves, rational and/or numbers:

    For a hundred and fifty years, in the pasture of dead horses,
    roots of pine trees pushed through the pale curves of your ribs,
    yellow blossoms flourished above you in autumn, and in winter
    frost heaved your bones in the ground—old toilers, soil makers:
    O Roger, Mackerel, Riley, Ned, Nellie, Chester, Lady Ghost.
    Donald Hall (b. 1928)

    While the miser is merely a capitalist gone mad, the capitalist is a rational miser.
    Karl Marx (1818–1883)

    The forward Youth that would appear
    Must now forsake his Muses dear,
    Nor in the Shadows sing
    His Numbers languishing.
    Andrew Marvell (1621–1678)