Just as imaginary numbers are complex number, there beginnings are also complex. Although Italian Renaissance mathematician Gerolamo Cardano is credited with the first acknowledgement of the existence of imaginary numbers in his book Ars Magna published in 1545, the topic is somewhat controversial. Niccolo Fontana Tartaglia communicated some of the information in Cardano’s book which created a ten year long dispute between the two men since Cardano had agreed not to publish the results. Cardano’s student Lodovico Ferrari also solved problems worked out in Cardano’s book. Because of the above two factors, it is often thought that Cardano received helped with the concept of imaginary numbers and may not have been their actual discoverer.
Imaginary numbers slowly became more recognizable. Greek mathematician and engineer Heron of Alexandria is considered the first to have actually observed these numbers and Rafael Bombelli was the first person to establish a set of grounds rules for using multiplication in regard to imaginary numbers. Despite these achievements and rules, imaginary numbers were still considered to be fictitious math. The use of imaginary numbers didn’t first become widely accepted until their use by Swiss mathematician Leonhard Euler and Norwegian Surveyor Caspar Wessel who established their significance to the field of geometry regarding planes.
Imaginary numbers weren’t particularly necessary in earlier applications of math because in algebra the roots of quadratics that came up as complex in their requirements of imaginary numbers were just discarded. The main reason that imaginary number truly became necessary to the field of math was to reconcile the differences in the proposed exact roots of cubic polynomials where it was required to take square roots of negative numbers.
Despite gaining acknowledgement in Cardano’s book, as we see in the above paragraph, imaginary numbers were only acknowledged in concept because of their lack of necessity at that period of time. The actual name ‘Æ'”¹ — “imaginary numbers’ came about during the 17th century and was meant to be offensive because the concept was considered useless. Although tired college students may disagree at times, despite the name remaining the same, today the feelings towards imaginary numbers are vastly different. Imaginary numbers find themselves praised for their use in a variety of practical applications in fields such as physics and engineering. Some of these applications include monetary tracking, cartography, vibration analysis and quantum mechanics.
As stated above, imaginary numbers are a part of the complex number system. They do not have a position on the number line. Imaginary numbers are normally denoted by the lowercase letter i and have the form bi where b is any real number and bi is called the pure imaginary number. When added to any real number a, it forms a complex number in the form a+bi. Their special property is that the square of an imaginary number results in a negative number. An example of this is i squared equals negative Because part of this complex form is real and part of it is imaginary, imaginary numbers can be graphed but not on the traditional one plane number lines that pure real numbers can. In order to appropriately graph imaginary numbers, a two-dimensional plane is required. In this instance, the traditional x-axis is used to graph the real number portion of the complex number, the a. The traditional y-axis is used the graph the imaginary portion of the form, the bi. Pure imaginary numbers are on the vertical axis of this plane except for at the point of origin, (0,0).
The history of imaginary numbers shows us that although something may just start out as an idea, that given the right amount of attention it can turn into something that is frequently used and as very realistic and practical uses especially in the use of scientific and real life applications. Although given the name imaginary numbers during the 17th century as a derogatory term, the name stuck and they’ve time and time again since then proven that they are indeed very real.
