THE HISTORY...
WHY THE SQUARE ROOT OF A NEGATIVE NUMBER SHOULDN'T EXIST...
People used the laws for positive numbers to explain negative numbers. Using the distributive law, we can see what \((-1)(-1)\) is: $$0=(-1)\bullet 0$$ because \(a\bullet 0=0\) for any \(a\), $$=(-1)(1+(-1))$$ because \(0=1+(-1)\), $$=(-1)\bullet 1+(-1)(-1)$$ by the distributive law, $$=-1+(-1)(-1)$$ because \(a\bullet 1=1\) for any \(a\),
hence $$1=(-1)(-1)$$ Therefore, we see why \((-a)^2=a^2\) for any positive number \(a\) and consequently that when we square any number, positive or negative, we come up with a positive number. Hence, the square root of a negative number should not exist. Nevertheless, we take the root of negative numbers and therefore specify between real and imaginary numbers to distinguish between roots of numbers that fall on the number line and those which do not. (Stillwell, 27)

Early mathematicians thought of the square root as a side because of Pythagoreans Theorem. Italians in the 1500s would refer to the square root of a number as its "lato" which is the word for side. Thus, it would make sense that the square root of a negative number would not exist (since there cannot be a negative side). The square root of negative numbers was used in mathematics for over three hundred years before a geometric understanding of it was actually discovered. (Mazur, pg 9)

THE CUBIC EQUATION...
Finding a solution for the cubic and the biquadratic equations was "perhaps the greatest contribution of algebra since the work of the Babylonians some 3000 years earlier." (Burton, pg 289) For several hundreds of years people searched for a cubic formula which could be used to solve a cubic equation like that of the quadratic formula which allows people to easily solve a quadratic equation. One of the first to find such a formula was Scipione del Ferro of the University of Bologna, who was able to solve the cubic equation for the special case \(x^3+px=q\) where p and q are positive. At that time period, methods or proof of mathematical discoveries were not commonly shared. This was due to the fact that "scholarly reputation was largely based on public contests." (Burton, pg 290) Thus, del Ferro didn't share his work with many. Nevertheless, he did share his secret with his pupil Antonio Maria Fiore.

In 1530 Nicolo Tartaglia was sent two problems to solve: \(x^3+3x^2=5\) and \(x^3+6x^2+8x=1000\). After he successfully solved both, he claimed that he could solve any cubic equation in the form of \(x^3+px=q\). Antonio Fiore, who had del Ferro's secret, challenged Tartaglia to a public problem-solving contest not believing Tartaglia's claim. The contest was comprised of 30 problems proposed by each contestant and it was decided that whoever could solve the most problems in 50 days would be the winner. Tartaglia knew that Fiore had been told del Ferro's secret and worked hard to find a general formula in preparation of the contest. His hard work must have paid off because he found a scheme for solving cubic equations for cubic equations missing the second-degree term and ended up solving the proposed problems in a matter of two hours. Fiore on the other hand, failed to solve even one problem.


Now Girolamo Cardano heard of the events and begged Nicolo Tartaglia to tell him the solution to the cubic equation. He told him he would publish Tartaglia's work in his next book Practica Arithmeticae , but Tartaglia turned down the offer. Tartaglia said that no one would look at the footnotes to see it was actually his work and that he could publish it himself. However, Cardano didn't give up. He invited Tartaglia to visit him and after "many entreaties and much flattery" (Burton, pg 293) Tartaglia told him his method. He made Cardano promise that he would not publish his work without giving him the credit. Yet, rumors were spread that Tartaglia was not the first to come up with such a general formula for the cubic, and after Cardano verified these claims he decided the promise was void. Consequently, Cardan published the formula and method of proof in his next work Ars Magna in 1545 reporting that he had gotten the solution from Tartaglia, but that he had carried out the proof himself. Thus, Cardano received Tartaglia's fame (although Tartaglia had only given Cardano the formula for \(x^3+px=q\) Cardano did solve for all of the other forms).

GIRALAMO CARDANO...

In the mid 1500s, Western algebraists did not accept negative numbers. One could find negative numbers in European an Arabic texts; however, those like the Western algebraists prefered to write equations so that only positive terms appeared. Thus, mathematicians had "confined their attention" (Burton, pg 294) to work with roots of equations which were positive numbers (The History of Mathematics). Because equations were written this way, Girolamo Cardano came up with the general formula for the cubic, \(x=\sqrt[3]{\frac{q}{2}+\sqrt{\frac{q^2}{4}-\frac{p^3}{27}}}+\sqrt[3]{\frac{q}{2}-\sqrt{\frac{q^2}{4}-\frac{p^3}{27}}}\). There was one problem with this form. When \((\frac{q}{2})^2<(\frac{p}{3})^3\) he would obtain square roots of negative numbers. Therefore, Cardano was the first to note negative roots of equations as well as that a cubic might have three roots.

While negative roots had been obtained while finding solutions to the quadratic formula, these solutions had been thrown out up until this point. Even though Cardon did notice such things as negative roots, he tried to keep them out of his work. Nevertheless, in Ars Magna he used the expressions \(5 + \sqrt{-15}\) and \(5 - \sqrt{-15}\) as solutions to the quadratic equation \(x(10-x)=40\). Cardon seemed torn with this result. He made the comment, "Putting aside the mental tortures involved , multiply \(5 + \sqrt{-15}\) by \(5 - \sqrt{-15}\), making \(25 - (-15)\), whence the product is 40...So progresses arithmetic subtlety the end of which, as is said, is as refined as it is useless." (Burton, pg 295)

RAFAEL BOMBELLI...
"Rafael Bombelli was the first mathematician bold enough to accept the existence of imaginary numbers." (Burton, pg 298) In his publication, Algebra, in 1572, he solved the equation \(x^3=15x+4\). By plugging in the values of p and q, Bombelli obtained \(x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}\). Bombelli was now working with negative roots or imaginary numbers. He assumed that he could calculate with complex numbers just as he could with real numbers and carried out this process of solving. Therefore, he moved in and out of the complex domain, ending with the all real solutions \(4, -2+\sqrt{3}\) and \(-2-\sqrt{3}\). Therefore, he proved Cardona's formula valid and exemplified that real number solutions could be found in working with imaginary numbers.

Because of Bombelli's success in calculating with complex numbers, people saw that imaginary numbers could be useful. However, imaginary numbers were fully accepted as numbers until the 1800s.

RENE DESCARTES...
In Rene Descartes last book of the Geometrie , he focuses on algebra and equations. He suggests that all of the terms of an equation is brought to one side and set equal to zero. He wasn't the first to suggest this, but he was the first to notice the benefits of doing so. He realized that "a polynomial \(f(x)\) was divisible by \((x-a)\) if and only if \(a\) was a root of \(f(x)\). We find too an intuitive proof of the theorem that any equation of degree \(n\) has \(n\) roots." (Burton, pg 337) Therefore, he opened up the idea of imaginary roots.

Descartes rule of signs is still used today and allows one to look at the signs and put an upper bound on the number of positive roots. This rule is stated as: "The number of positive roots (each root counted as often as its multiplicity) of an equation $$f(x)=a_0x^n+a_1x^{n-1}+...+a_{n-1}x+a_n=0, \hspace{.25 in} a_0>0 $$ with real coefficients is equal either to the number of variations in the signs of its coefficients or to this number decreased by a positive even integer." (Burton, pg 338) Descartes came up with a similar rule for the number of negative roots stating: "No equation can have more negative roots than there are variations of sign in the coefficients of the polynomial f(-x)." (Burton, pg 338)

If the largest possible number of positive roots and the largest possible number of negative roots gave a sum which was less than the degree, or the number of roots, of a polynomial, then Descartes noted the existence of imaginary roots.

IMAGINARY NUMBERS IN A MORE MODERN TIME...

The complex plane which is made up of an x-axis describing the real numbers and a y-axis which describes imaginary numbers was first considered by John Wallis. Caspar Wessel also considered this idea, but it wasn't until Carl Friedrich Gauss suggested this idea in 1831 that it became more accepted.

In 1777, Euler came up with the notation of \(i\) to represent \(\sqrt{-1}\). Euler also came up with what is known as the most beautiful mathematical equation of $$e^i\pi$$.

Today imaginary numbers are used in hundreds of different things from the flow of fluid to saving pictures on a memory card.