history of mathematics - National STEM Centre

Descartes
Today we call these
numbers complex
numbers. The current
notation for -23 s
. The character
i indicates -- A
complex number is thus
composed of a real part,
here •£, and an imaginary
part,
106
Activity 8.8 The rule of signs
Apparently, Descartes also considers the solutions of x2 - x + 6 = 0 to be positive
solutions: \ + \•^p23 and ^--^-V^23 .
In La Geometrie these so called imaginary numbers are mentioned explicitly.
38 Neither the true nor the false roots are always real; sometimes they are
imaginary; that is, while we can always conceive of as many roots for each
equation as I have already assigned, yet there is not always a definite
quantity corresponding to each root so conceived of. Thus, while we may
conceive of the equation x 3 -6x 2 + 13*-10 = 0 as having three roots,
yet there is only one real root, 2, while the other two, however we may
increase, diminish, or multiply them in accordance with the rules just laid
down, remain always imaginary.
From Descartes's statement that the number of solutions of an equation is equal to
its order, you can conclude that he accepted imaginary solutions of equations.
1 The equation x 4 - 5x 2 + 4 can be factorised into (x - \)(x + l)(x - 2)(x + 2).
Can the sign rule be applied to x 4 - 5x 2 +4 = 0?
2 In the previous chapters, the trisection of an angle resulted in the equation
Z 3 =3z-q, with q > 0. How many positive solutions has this equation?
3 The construction of solutions of the equation z 3 = 3z - q was carried out in
Chapter 7, Activity 7.15. The solutions are the x-values of g, G and F. Which of
these three solutions is negative?
Descartes's ideas about equations provides a tool for reducing algebraic equations to
their simplest form. This nearly completes his attempts to solve geometrical
construction problems in a generally applicable method. He ends his book with
examples of constructions, among which is the trisection of an angle.
All in all, La Geometrie is a book full of revolutionary ideas, even though it was
intended to tackle a classical goal, namely, solving geometrical construction
problems. The book had a great effect on mathematics: it caused the birth of
analytical geometry and the appearance of differential calculus. Descartes's wish,
that future generations would appreciate his work, has come true.
39 But it is not my purpose to write a large book. I am trying rather to
include much in a few words, as will perhaps be inferred from what I have
done, if it is considered that, while reducing to a single construction all the
problems of one class, I have atthe same time given a method of
transforming them into an infinity of others, and thus of solving each in an
infinite number of ways; that, furthermore, having constructed all plane
problems by the cutting of a circle by a straight line, and all solid problems
by the cutting of a circle by a parabola; and, finally, all that are but one
degree more complex by cutting a circle by a curve but one degree higher
than the parabola, it is only necessary to follow the same general method