Algebra I
April 2014
Square Roots of Negative
Numbers?
The most commonly occurring application problems that
require people to take square roots of numbers are
problems which result in a quadratic equation.
ex. Determine the dimensions of a rectangle with a
perimeter of 9 cm and an area of 5 cm2.
ex. Determine the dimensions of a rectangle with a
perimeter of 5 cm and an area of 2 cm2.
Thus, having to take the square root of a negative number
in this context means that such a rectangle does not exist.
Motivation: The Cubic Formula
The first people who worked on the cubic formula lived in
Persia (modern day Iran).
Omar Khayyám (1048-1131)
Sharaf al-Dīn al-Tūsī (1135-1213)
Both of these mathematicians discovered partial solutions,
but did not fully develop the cubic formula.
Motivation: The Cubic Formula
From about 1150 to about 1600, in Europe, especially
modern day Italy, it was common to have mathematical
competitions in which someone would give equations for
competitors to solve.
The winner of these competitions received a monetary
award.
Motivation: The Cubic Formula
Scipione del Ferro discovered a partial solution to the
cubic formula. His solution worked for equations of the
form
x3 + mx = n
for positive integers m and n.
He kept his solution secret until just before he died, when
he shared his solution with Antonio Fiore in 1526.
Motivation: The Cubic Formula
In 1530, Niccolò Tartaglia challenged Antonio Fiore to a
mathematics competition.
The equations that Fiore had to solve were of the form
x3 + mx2 = n
instead of
x3 + mx = n.
Tartaglia won the competition.
Motivation: The Cubic Formula
Niccolò Tartaglia had worked on a system for solving
cubic equations. After his victory in 1530, he developed it
further.
Gerolamo Cardano was interested in mathematics, but not
the competitions. When he realized that Tartaglia must
have a system for solving certain types of cubic equations,
Cardano asked him for the method.
In 1539, Tartaglia shared only a portion of his knowledge
of the cubic formula (in the form of a poem) with Cardano
on the condition that he keep it secret.
Motivation: The Cubic Formula
Gerolamo Cardano and one of his students, Lodovico Ferrari,
work on the other cases to which Niccolò Tartaglia did not
provide a solution. They solve all of the cases and thus
completely discover the cubic formula.
In 1545, Cardano discovers Scipione del Ferro’s prior work
with cubic equations…
…and felt justified in publishing his complete solution in the
book Ars Magnæ, Sive de Regulis Algebraicis (The Great Art,
or the Rules of Algebra).
Later that year, Tartaglia takes part in a mathematics
competition that Ferrari wins. He then realizes that the secret
is out.
The Cubic Formula in Action
Solve x3 – 7x + 6 = 0.
Plugging into the cubic formula yields
.
It turns out that this solution is x = 2.
Complex Numbers: A Whole New
World
Even after the discovery of the cubic formula, it took
some time for mathematicians to accept and understand
complex numbers.
René Descartes’ work, especially on the Cartesian
coordinate system, sped up the process of mathematicians
learning and understanding complex numbers.
Are There More Numbers?
In 1799, Carl Friedrich Gauss proved the Fundamental
Theorem of Algebra.
Every nonconstant polynomial with complex
coefficients has a complex root.
In other words, for the purpose of solving polynomial
equations, there is no need to hunt for more new numbers.
Are There More Formulas?
Lodovico Ferrari discovered the quartic formula in 1540.
It appeared in Gerolamo Cardano’s Ars Magna in 1545.
In 1799, Paolo Ruffini wrote a five hundred page proof
attempting to show that the quintic formula does not exist.
His contemporaries did not accept his proof.
In 1824, Niels Abel proved that the quintic formula does
not exist. His proof, which was only six pages long, is
correct.
In 1832, Évariste Galois found and proved the conditions
under which a polynomial equation is solvable by radicals.
Further Reading
Katscher, Friedrich. (2011). How Tartaglia Solved the
Cubic Equation. In The Mathematical Association of
America (MAA) Mathematical Sciences Digital Library.
Retrieved April 24, 2011 from
http://www.maa.org/publications/periodicals/convergence/
how-tartaglia-solved-the-cubic-equation-tartagliassolution
Rotman, Joseph. (1995). A First Course in Abstract
Algebra. Upper Saddle River, NJ: Prentice Hall.
Rotman, Joseph. (1998). Galois Theory. New York, NY:
Springer.