From Cubic Equations to the Electric Chair

You probably remember studying quadratic equations in school. These are the equations which go like this,

ax2 + bx + c = 0

You probably remember this is a neat little formula to find the 2 possible values of x. Don’t worry, I’m not going to ask you to recall it. But there is another type of equation called the cubic which goes like this,

ax3 + bx2 + cx + d = 0

These are much trickier to solve. Many classical and medieval mathematicians (including Bhaskara in India, Fibonacci in Italy and Omar Khayyam in Persia) bashed their skulls against this problem till it was finally solved in the 16th century by the Italian mathematician Scipione del Ferro. Ferro himself thought that he had only solved a special category of cubic equations. He didn’t realize that his solution could be extended to include all cubic equations because negative numbers were unknown at the time.

One problem was soon realized. These solutions required the use of strange new numbers. For instance, it was known that

11 X 11 = 121


(-11) X (-11) = 121

But what did it mean when we asked for the square root of -121? The idea that something could multiply with itself to yield a negative number made no sense. Today, of course we know them as imaginary numbers and learn them in school denoting them by the symbol i. But at the time, they were a nuisance. They kept appearing in the calculation of mathematicians. In fact, even when the final solution to the cubic equation might be a simple number, the steps to get it wold be riddled with such imaginary numbers.

11i X 11i = -121

Rafael Bombelli first systematized the rules of dealing with such numbers and Rene Descartes coined the name, calling them imaginary numbers. There the situation remained for centuries. Imaginary numbers were a mathematical curiosity which appeared along the way to other solutions in mathematics till 1886.

At this time, a number of botched hangings in the US created a mounting criticism of this procedure for carrying out capital punishment. A three member commission was created by the New York State governor to find a more humane form of execution. Among the people they spoke to was Thomas Edison.


Thomas Edison

At the time, Edison was involved in the current wars. Edison had pioneered the use of direct current (DC) machines and had numerous patents on machines working with DC. However, a new form of electric current had come about championed by Nicola Tesla and Westinghouse called alternating current (AC) where the direction of current kept changing. AC enjoyed a key advantages over DC in that it was more efficient to distribute over long distances. Unfortunately, the mathematics required to understand the working of AC was very complicated.

Edison tried to demonize the new technology. He suggested that the commission should consider execution by means of an electric chair using a Westinghouse AC generator.


Electric chair

Edison even got his employees to build such an electric chair. The chair was built by Arthur Kenelly. Ironically, during his working with AC, Kenelly realized that the mathematics used to describe this current would be greatly simplified if he made use of imaginary numbers. By this time, imaginary numbers had made their way into textbooks.

The simplicity introduced by using imaginary numbers to describe the phase of the alternating current contributed greatly to its understanding and acceptance among engineers. Kenelly left Edison’s lab to start his own electrical engineering firm and wrote treatises on alternating electric current and introduced the idea of using imaginary and complex numbers into electrical engineering where they are used to this day in the teaching end practices of electrical engineering.








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