Cardano formulaGirolamo Cardano , is the solution formula of the cubic equation, which gives the cubic equation xthree+The three solutions of px+q=0 are xone=u+v，xtwo=uw+vwtwo，xthree=uwtwo+vw。Since the general cubic equation ythree+aytwo+By+c=0 can be transformed into the form of x after the substitution of unknown quantity y=x-a/3three+The cubic equation of px+q=0.Therefore, the cubic equation with any complex coefficient can be solved by using the Kaldano formula, which is actuallyTartaglia(TN. artaglia) first discovered it in 1541, but it was not published publicly, but told G. Cardano under the promise of confidentiality. The latter reported this result in 1545Published in his book Dashu[3]And later people called it Kaldano formula[1]。
Kaldano formula is a famous formula for finding roots, which refers to the cubic equation with real coefficients
The root formula of x=α+β, where
And α β=- p/3, this formula can also be applied to the complex coefficient cubic equation[2]。
Related introduction
Announce
edit
Italian mathematicianCaldano(G. Cardano) first published the above formula in the book "Dashu" published in 1545, which came from Italian mathematiciansTartaglia(N. Tartaglia), but Cardano gave a geometric proof of the formula.
When p and q are real numbers
Is the discriminant of equation (1).
When D>0, equation (1) has three different real roots, which is called irreducible case;
When D=0, equation (1) has three real roots, when p and q are not zero, there are two multiple roots and a single root;
When D<0, equation (1) has one real root and two conjugate imaginary roots[2]。
The Kaldano formula shows that the cubic equation has a radical solution. His student, L. Ferrari, obtained the radical solution of the quaternion equation with one variable by reducing the order, which triggered people's research on the radical solution of algebraic equations of more than five times, and promoted the generation and development of mathematics in recent generations.In addition, because of the appearance of using imaginary numbers to represent real roots in irreducible cases, people once again encounter the square root of negative numbers, thus promoting the understanding of the rationality of imaginary numbers.In 1572, the Italian mathematician Bombelli, R., discussed how to solve the unary cubic equation x in his book Algebrathree=15x+4, its three roots are 4,
。But the application of Kaldano formula is
After studying, Bonbury believed that the square root of negative numbers should be operated like "ordinary numbers".Later, the German mathematician Leibniz (G.W.) also studied the irreducible situation, and was convinced that it was impossible to solve this situation without imaginary numbers by algebraic methods.This makes people gradually realize that the square root of negative numbers has a certain objective basis and rationality, and speeds up the recognition process of people's acceptance of imaginary numbers.French mathematicianWeida(F. Viete) In On the Identification and Revision of Equations (completed in 1591 and published in 1615), he used trigonometric identities to give the root of equation (1) in the irreducible case as
