The Value of Imaginary Numbers. Most of us are familiar with the numbers that exist on a one-dimensional scale called the number line, as

Transcription

1 Paige Girardi Girardi 1 Professor Yolande Petersen Math 101, MW 11:40-1:05 April The Value of Imaginary Numbers Most of us are familiar with the numbers that exist on a one-dimensional scale called the number line, as illustrated in figure 1.1. On this number line, we have familiar numbers such as 0,1,2, fractions, negative numbers, and irrational numbers such as root 2. However, without imaginary numbers, our system of numbers is incomplete. So, what are imaginary numbers? An imaginary number is a number that can be shown as a real number multiplied by i. Zero and all positive and negative numbers are real numbers. As we can also see in figure 1.1, imaginary numbers exist in another dimension of the number line. This dimension holds problems such as the square root of negative one. However, for hundreds years, such problems were mathematically considered to be fictitious or even impossible. Figure 1.1 Professor of Mathematics at the University of Washington, Robert E. Moritz stated in his book Elements of Plane Trigonometry (with five-place tables) that choosing to call these numbers imaginary suggests that these numbers aren t real. Moritz mentions that mathematician Carl Friedrich Gauss, who extensively studied imaginary numbers, also held this belief. Gauss thought the name imaginary shrouded these numbers in

2 Girardi 2 obscurity and only enhanced the world of mathematics confusion and disbelief in their use. Gauss believed this so much so that he thought imaginary numbers should be renamed as lateral numbers (Moritz, 1915). As one can see, there has been some confusion and controversy concerning imaginary numbers. Several centuries ago, mathematicians not only ignored the validity of imaginary numbers and even negative numbers, but they would reconfigure their equations in order to avoid these numbers. In the 14 th century, an Italian Mathematician Scipione del Ferro made a discovery that influenced later mathematician, Rafael Bombelli to define and make sense of imaginary numbers. Ferro was finding a formula for equations with the highest power of 3. As illustrated in figure 1.2, we see a problem that Ferro recreated to have a negative last term and no x squared using imaginary numbers. He ended up writing it as x cubed plus c x is equal to d, requiring c and d to be positive. To complete the problem, x had to be on one side and all the constants had to be on the other side (Merino, 2006). Ferro kept his formula with imaginary numbers a secret until his death in Figure 1.2 Mathematician Gerolamo Cardano discovered Ferro s work, which included these three problems: Cardano published a book Ars Magna in 1545, with these three cases. What Cardano found was that the second problem presented the difficulty of having a square root of a negative number. Cardano s idea was to substitute x=u +v into x 3 = px+q to get the fallowing:

3 Girardi 3 However, Cardano did not discuss this case much or in depth in Ars Magna because he had misgivings about it. It was later Rafael Bombelli who addressed this case and defined imaginary numbers (Merino, 2006). In 1572, Rafael Bombelli defined imaginary numbers in his three books that were under the name, L Algebra. Bombelli solved equations using Ferro s method. In L Algebra, Bombelli was the first European to explain how computations with negative numbers are performed. Bombelli also included in his book the application of complex numbers. Bombelli showed that complex and imaginary numbers had real solutions that could be obtained from Ferro s formula for the answer to a cubic, even when it involved the square root of a negative number (O Connor, 1999). Bombelli believed that people were confused about imaginary numbers simply because they tried to compute them the same as they would real numbers. He made it clear that imaginary numbers have their own rules of arithmetic. To show this, Bombelli gave a specific name to square roots of negative numbers, which made them distinct from regular radicals. He called the imaginary number i for plus of minus and i for minus of minus. Bombelli was able to show that imaginary numbers were important for solving quadratic and cubic equations. In his book, Bombelli explains the multiplication of real and imaginary numbers, as illustrated in figure 1.3 (O Connor, 2000).

4 Girardi 4 Figure 1.3 Since Bombelli s contribution on the use of imaginary numbers, mathematicians now understand and appreciate the value of imaginary numbers. This then prompted them to advocate for imaginary numbers to be accepted and used in mathematics. Several mathematicians of the 19 th century, such as William Rowan Hamilton, developed ways of representing imaginary numbers as pairs of real numbers so that they would be less confusing and more believable. Now, imaginary numbers are a fundamental part of mathematics and are used to answer a common algebra problem presented to most high school and college students: the solving of an equation with the quadratic formula. For example, in an equation like x 2-4x + 29= 0, we want to solve with the quadratic formula. First, we plug this problem into the quadratic formula. By doing so, our equation will look like this: Next, we solve what is inside the radical, which is -100.

5 Girardi 5 Now we are dealing with a negative number s square root. We use i to represent that negative square root, which comes out to be 10i. Next, we factor the equation by two. This gives us, inside the parenthesis, 2 plus or minus 5i. We then cancel out the 2s. Now we have our two answers, as shown below. Now, imaginary numbers are practical and familiar numbers we learn about and use in algebra. They have many uses, as a result of many mathematicians struggling to believe in them and others striving to prove their existence. Ever since Rafael Bombelli defined the imaginary number i, we are able to solve problems like the square root of a negative number. We all are able to use these special numbers to solve problems that were once considered to be invalid or have no solution.

6 Works Cited Girardi 6 Merino, Orlando. "A Short History of Complex Numbers." Math.uri.edu. N.p., Jan Web. 7 Apr Moritz, Robert E. Elements of Plane Trigonometry (with Five-place Tables): A Text-book for High Schools. New York: Wiley, Web. 03 Apr O'Connor, JJ. "Rafael Bombelli." Bombelli Biography. N.p., Jan Web. 03 Apr O'Connor, JJ. "Scipione Del Ferro." Ferro Biography. N.p., July Web. 03 Apr