Let's Get Real Here

The imaginary numbers were first discovered by Girolamo Cardano who lived during the Renaissance (1501-1576). He was a physician, philosopher, mathematician, astrologer, and prolific writer. As he spent most of his youth gambling, he was able to develop many cognitive and problem solving strategies. He obtained a Doctorate in medicine from Padua, however his study of medicine came to an end when he wasn't accepted into the College of of Physicians. He then devote his time to his greatest work, the Ars Magna (The Great Art). There was only one instance where he included his discovery of imaginary numbers. By solving the equation x(10-x)=40, he obtained the roots 5+√-15 and 5-√-15. However, he noted these roots as useless.

Within his work, he also included his discoveries from working with the cubic equation x3+px=q. When attempting to solve it, he comes across imaginary numbers once again and dismisses them.So, in order to avoid imaginary numbers, many mathematicians wrote their equations with only positive terms. Rafael Bombelli was the first mathematician that took particular attention to the imaginary numbers. He worked a lot with the Ars Magna. He was intrigued by the idea of taking the square root of a negative number. Although his discoveries didn't lead to a full understanding and acceptance of imaginary numbers, it brought them out of the dark, scary abyss and into the light.

Girolamo Cardano                      Rafael Bombelli

                   



Next to come along was Rene Descartes (1596-1650). In his last book of the Geometrie, he he worked mainly with algebra. By taking all terms of an equation to one side and setting it equal to zero, he discovered many properties concerning the polynomial's roots. If a is a root of f(x), then the corresponding linear polynomial (x-a) is a factor of f(x). In his work, he also provides a proof of his theorem that states, "Any equation of degree n has n roots." Descartes also became very well known for his rule of signs. Given an equation:

f(x)=aoxn+a1xn-1+...+an-1x+an=0, ao>0,


the number of positive real roots is less than or equal to the number of sign changes. Similarly, the number of negative real roots is less than or equal to the number of sign changes in f(-x). However, since the sum of the number of positive and negative real roots isn't always equal to n, this this gave rise to the existence of imaginary roots.

Now have the ability to graph complex numbers. The y-axis becomes the imaginary axis, and the x-axis is the real axis. This is called the complex plane.

Mathematicians in today's world are familiar with i as the representation for √-1. This notation was introduced by Leonhard Euler in the 1700s. Another neat property of imaginary numbers is that:

i0=1

i1=i

i2=-1

i3=-i

i4=1

i5=i

i6=-1

i7=-i

And this pattern continues to repeat. So, in=in(mod 4).