Wednesday, September 4, 2019
Imaginary and complex numbers
Imaginary and complex numbers When Are We Ever Going to Use This? Ã¢â¬â Imaginary and Complex Numbers The number Ã¢ËÅ¡-9 may seem impossible, and it is when talking about real numbers. The reason is that when a number is squared, the product is never negative. However, in mathematics, and in daily life for that matter, numbers like these are used in abundance. Mathematicians need a way to incorporate numbers like Ã¢ËÅ¡-9 into equations, so that these equations can be solvable. At first the going was tough, but as the topic gained more momentum, mathematicians found a way to solve what their predecessors deemed impossible with the use of a simple letter i, and today it is used in a plethora of ways. History of Imaginary Numbers During the early days of human mathematical history, when someone reached a point in a equation that contained the square root of a negative number, they froze. One of the first recorded instances of this was in 50 AD, when Heron of Alexandria was examining the volume of a truncated pyramid. Unfortunately for him, he came upon the expression which computes to . However, at his time, not even negative numbers were Ã¢â¬Å"discoveredÃ¢â¬ or used, so he just ignored the negative symbol and continued on with his work. Thus, this first encounter with complex numbers was unsuccessful. It is not until the sixteenth century when the dilemma of complex numbers returns, when mathematicians attempt to solve cubic and other equations of higher-order. The Italian algebraist Scipione dal Ferro soon encountered these imaginary numbers when solving higher degree polynomials, and he said that finding the solution to these numbers was Ã¢â¬Å"impossibleÃ¢â¬ . However, Girolamo Cardano, also Italian, gave this subject some hope. During his mathematical career, he opened up the realm of negative numbers, and soon began analyzing their square roots. Although he admitted that imaginary numbers were pretty much useless, he shed some light on the subject. Fortunately, this little bit of light would soon turn into a full beam. In 1560, the Bolognese mathematician Rafael Bombelli discovered a unique property of imaginary numbers. He found that, although the number Ã¢ËÅ¡-1 is irrational and non-real, when multiplied by itself (squared), it produces both a rational and real number in -1. Using this idea, he also came up with the process of conjugation, which is where two similar complex numbers are multiplied together to get rid of the imaginary numbers and radicals. In the standard a+bi form, a+bi and a-bi are conjugates of each other. At this point, many other mathematicians were attempting to solve the elusive number of Ã¢ËÅ¡-1, and although there were many more failed attempts, there was a little bit of success. However, although I have been using the term imaginary throughout this paper, this term did not come to be until the 17th century. In 1637, Rene Descartes first used the word Ã¢â¬Å"imaginaryÃ¢â¬ as an adjective for these numbers, meaning that they were insolvable. Then, in the next century, Leonhard Euler finalized this term in his own Eulers identity where he uses the term ifor Ã¢ËÅ¡-1. He then connects Ã¢â¬Å"imaginaryÃ¢â¬ in a mathematical sense with the square root of a negative number when he wrote: Ã¢â¬Å"All such expressions as Ã¢ËÅ¡-1, Ã¢ËÅ¡-2 . . . are consequently impossible or imaginary numbers, for we may assert that they are neither nothing, not greater than nothing, nor less than nothing, which necessarily renders them imaginary or impossible.Ã¢â¬ Although Euler states that these numbers are impossible, he contributes with both the term Ã¢â¬Å"imaginaryÃ¢â¬ and the symbol for Ã¢ËÅ¡-1 as i. Although Euler does not solve an imaginary number, he creates a way to apply it to mathematics without much trouble. Throughout the years, there have been many skeptics of imaginary numbers; one is the Victorian mathematician Augustus De Morgan, who states that complex numbers are useless and absurd. There was a tug-of-war battle between those who believed in the existence of numbers such as i and those who did not. Soon after Rene Descartes contributions, the mathematician John Wallis produced a method for graphing complex numbers on a number plane. For real numbers, a horizontal number line is used, with numbers increasing in value as you move to the left. John Wallis added a vertical line to represent the imaginary numbers. This is called the complex number plane where the x-axis is named the real axis and the y-axis is named the imaginary axis. In this way, it became possible to plot complex numbers. However, John Wallis was ignored at this time, it took over a century and a few more mathematicians for this idea to accepted. The first one to agree with Wallis was Jean Robert Argand in 1806. He wrote the procedure that John Wallis invented for graphing complex numbers on a number plane. The person who made this idea widespread was Carl Friedrich Gauss when he introduced it to many people. He also made popular the use of the term complex number to represent the a+bi form. These methods made c omplex numbers more understandable. Throughout the 1800s, many mathematicians have contributed to the validity of complex numbers. Some names, to name a few, are Karl Weierstrass, Richard Dedekind, and Henri Poincare, and they all contributed by studying the overall theory of complex numbers. Today, complex numbers are accepted by most mathematicians, and are easily used in algebraic equations.