Impacts of the Imaginary Number on Mathematics

Mathematics was man’s first approach to understanding the world around them since the beginning of humanity. The study grew with history in various forms with every human civilization, and as time passed, more discoveries were made that allowed humanity to reach great heights in agriculture, architecture, social structure, and their culture. Great mathematicians continued extensive studies and experiments with various values that existed in their time to further improve the study. However, the concept of the imaginary number i was developed fairly recently. This essay is written from the fascination of abstract mathematical concepts, to develop the impacts of the imaginary number on mathematics.

In order to research this topic, I am required to view numerous proposed and established claims of the imaginary number’s history, and find these ideas being used with real numbers to obtain solutions to problems we have today in other subjects such as physics, and astronomy. The purpose of this essay is to further research the significance of the imaginary number, i, and its contributions to modern mathematics, physics, engineering, and other sciences. The expansion of knowledge on this topic will further propel the study of mathematics in the future.

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Mathematics is the only subject that can explain the universe in a logical, unbiased, and truthful way. Mathematics has been in the roots of the development of advanced civilizations, in any time period. As humanity advanced, mathematics expanded. However, dilemmas were created as a consequence of its advancements. People created concepts within mathematics which a human brain could not fully understand. Concepts such as the imaginary number, i, are impossible to truly comprehend with our limited minds. However, the beauty of mathematics is that even the most impossible seeming, imaginary number, i has a history, and has significant impacts to modern mathematics.

In mathematics, a square number is defined as an integer that is the product of some integer with itself. For example, 9 is a square number, as it is the product of 3 3. This can be written in an alternate notation, 3², which is pronounced as “3 squared”. The name square comes from the fact that the area of a square is the product of its 2 equal side lengths. A square number is always a positive value, as positive positive = positive, and negative negative = positive also. If squaring exists as an operation, there has to be the counter operation; the square root, or . The square root takes a square number and reduces it to the single factor that was squared to form the square number. For example, = 3. As all square numbers are positive, square roots of negative numbers are illogical, or it was only considered illogical in the past…

= i, or the imaginary number, has the property of becoming a real number when raised to the power of an even number; i² = ()² = – 1, or; i⁴ = ()⁴ = 1. A real number include all of the rational numbers, as in it is a whole number, or has an ending decimal value, and all of the irrational numbers, which have unending decimal values. The characteristic that all 3 types of numbers have in common is that they can be represented in a number line, in some form. Unlike these real numbers, i has no way to be represented on a line.  Furthermore, i is not the only imaginary number; it is the “unit” imaginary number, used as a part of a complex number. A complex number is a combination of a real number and an imaginary number, taking the form of x + iy, where x and y are real numbers. For example, 12 – 5i is a complex number. However, when x = 0, leaving only iy, such as 16i, it is then called a purely imaginary number. In contrast, if y = 0 leaving only x, the complex number is then a real number. In this sense, all real numbers are actually just subsets of complex numbers.

In calculations, complex numbers are often paired with conjugates, which is defined as the binomial formed by negating the second term of a binomial, in the form of x ± yi; in relation to complex numbers, it is the complex number with the imaginary part having the opposite sign. For example, the conjugate of the complex number 12 – 5i is 12 + 5i. These conjugates functions to eliminate the imaginary numbers from the denominator of a complex fraction, by multiplying the numerator and the denominator by the appropriate conjugate. The conjugate always = 1, so it does not alter the value of any equation. For instance, in an equation such as

 

it can be simplified by multiplying (which equals 1) to it, resulting in

= = 

yielding a single complex number,

As shown, the imaginary number is not some abstract concept of virtually zero use; it can be applied to real mathematics as simply as such. However, the idea of an imaginary number was not widely accepted until relatively recently in history, in the last 2 centuries or so.

Before the concept of imaginary numbers were even conceived of, mathematics in the western world was restricted to geometry, led by the Ancient Greeks. The Algebra that modern mathematics is familiar with was invented by the Hindus, which was later translated and improved by the Arabs, spear-headed by Arab Mathematician Al-Khwarizmi(780-850). At the time, however, the solutions to polynomials were restricted to positive solutions, omitting any negative quantities. Al-Khwarizmi’s algebra was then translated from Arab to Latin by Gerardus Cremonensis, and Leonardo Bonacci, also known as Fibonacci. (MerinoOrlando)

The first recorded use of complex numbers in seen in the works by Gerolamo Cardano. Cardano was an Italian mathematician during the 16^th century Renaissance. In fact, he is recognized as one of the most influential mathematicians of the time, being a prominent member for the foundation of probability, binomial coefficients, and binomial theorems. He also contributed to the invention of the combination lock, and the modern gyroscope. He published over 200 works over the course of his lifetime. One of his famous works, the “Ars Magna”, published in 1545, included the problem “To divide 10 in two parts, the product of which is 40”, or finding the solution to

– 10 + 40 = 0. (BogomolnyAlexander, Remarks on the History of Complex Numbers)

Cardano usually used geometric algebra in order to avoid any use of negative numbers by considering several different forms of quadratic equations; however, he decided to solve the question he declares impossible. He first divided 10 in half, making each 5. Then according to the methods he discussed in the previous section of his book, he squares 5, and subtracts 40 from it, leaving −15. He then square roots -15, which he then adds and subtracts from 5, leaving him with the roots (5 + ) and (5 − ). In mathematical terms, his operation was   

5² = 25

25 – 40 = -15

5 ±

(5 + ) (5 − ) = 40.

This is confirmed by simply multiplying the binomials:

(25 – 5 + 5 – −15)

=(25 + 15) = 40.

However, Cardano writes that in conclusion, this solution is useless, as it cannot be performed. (MerinoOrlando)

The next significant milestone was achieved by the mathematician Rafael Bombelli in his (1572) work, Algebra. He was the first to recognize the significance of √−1, and notates it “piú di meno”, or “plus of minus” in Italian. Bombelli was far more familiar with the operation of negative numbers than Cardano, and establishes the rules when handling different signed numbers. His works are as follows; the following is directly translated from his work in Italian:

Plus times plus makes plus (1 1 = 1)

Minus times minus makes plus ( – 1 – 1 = 1 )

Plus times minus makes minus ( 1 – 1 = – 1 )

Minus times plus makes minus. ( – 1 1 = – 1 )

He then annunciates the behavior of the number “plus of minus“:

Plus of minus times plus of minus makes minus ( = – 1 )

Plus of minus times minus of minus makes plus ( – = 1 )

Minus of minus times plus of minus makes plus ( – = 1 )

Minus of minus times minus of minus makes minus ( – – = – 1 ) (BogomolnyAlexander, Remarks on the History of Complex Numbers)

Bobelli took the same approach as other mathematician at the time when encountering negative roots as a solution to cubic and quadratic equations, often omitting them completely, or disregarding them. However, he did attempt once to solve a cubic using imaginary numbers, and succeeded, without realizing its validity.

The term “imaginary” was coined by the philosopher and mathematician René Descartes (1596-1650); he also coined the term “real number” to distinguish between real and imaginary roots of polynomials. He did not actually contribute to the mathematics aspect of i, but just provided a name for the poorly understood concept.

John Wallis (1616 -1703) was first to introduce a geometric interpretation of complex numbers, and believe that negative numbers were larger than infinity, but still less than 0. This thought was shared by the famous mathematician Leonhard Euler (1707 – 1783), who introduced the symbol i as the symbol for  , and “linked the exponential and trigonometric functions in the famous formula

eit = cos(t) + i ·sin(t).”

The geometric interpretation of complex numbers that modern mathematics agree with was first introduced by Caspar Wessel (1745-1818). Wessel treated complex numbers as vectors (which, he did not use the term “vector”), and derived most of their properties, including trigonometric form of multiplication (or, “algebraic” multiplication).

The acceptance of complex numbers in mathematical society was further elevated by Carl Friedrich Gauss (1777-1855) with the use of complex numbers to Number Theory. Gauss introduced the term “complex number”, which he defined as the combination of real and imaginary numbers. However, i was still not fully accepted and understood until the mid-19^th century, from the works of Sir William Hamilton, 9th Baronet, (1805-1865). He was responsible for the notation (x,y); he defined ordered pairs of real numbers of real numbers (a.b) to be a couple. This further implemented complex numbers as vectors or points on a plane, vector operators, and matrices. (MerinoOrlando)

As one can observe from the historical track of i, complex numbers were abstract concepts of little value to mathematics until the last two centuries; many, such as Cardano and Bombelli, disregarded i as a valid method for finding solutions. However, today, with a better understanding of complex numbers, we can now solve equations they weren’t able to solve for centuries, with proper explanations to support the answer.

With the knowledge of i, we are able to solve through some of the questions that the greatest mathematicians during the last few decades couldn’t solve. One of the problems was derived from the cubic formula, invented by the Mathematician Del Ferro (1465 – 1526). To solve a quadratic equation, or an equation having the form , means finding the values of x for when y =0. In other words, when the equation is graphed on a xy-coordinate graph, the x values of the points where the line crosses the x-axis. Conveniently, an Indian mathematician named Brahmagupta (597-668 AD) invented a “quadratic formula” in to facilitate the process of finding the solutions:

,

where terms a, b, and c correspond with the letters in . (KnaustHelmut)While this is not the quadratic formula we are accustomed to today, , it was still a revolutionary way to solve quadratics. Del Ferro aimed to create a formula for cubic equations that have the same level of convenience as the quadratic formula., and he succeeds. The formula looked something like this:

     , for the cubic equation in the form of  Cardano later acquired this secretly guarded formula and modified it to a much simpler form,

by using a change of variable x = to eliminate the x² value to form a simpler cubic equation, . Cardano published this formula in the previously mentioned Ars Magna (KnaustHelmut). However, Cardano faced a major problem; in a slightly different version of the equation, he found that his formula would break under certain circumstances: when < 0. For example, using a rather simple looking equation of

;

when plugged into Cardano’s extra modified formula,

= ,

The result involves a square root of negative numbers; these negative square roots were enough of a problem to cause Cardano to stop in his progress on this area. At the time, all negative roots as a solution was considered by mathematicians as the problem’s way of saying there are no solutions, and in most cases, it was true. Bombelli, however, while still not accepting the validity of the imaginary number, finished solving Cardano’s problem.  In the instance of a cubic, there has to be at least 1 real solution, because of the nature of the shape of a cubic on the xy- graph. At least 1 point had to cross the x- axis, at all circumstances. This is one of the “Fundamental Theorems of Algebra”; a polynomial function has to have n number of solutions for the largest nth power. Through testing some integers, Bombelli found that 4 is one of the solution to the equation:

4³ = 15(4) + 4

64 = 60 + 4

64 = 64

The solution, as anyone can see, is a real number; for this to be the case, Bombelli realized that the root of i parts of each half of the equation needs to “cancel out”, or equal to zero when added together, like this:

He then used this idea to form complex conjugates, and where and b are constants that we need to obtain, which we equate to each half of the equation:

We can then start solving for the constants by cubing both sides of the equation:

)³³

= )( +

=

     =

Now we need to separate the real and the imaginary parts:

 

and

Now since we know that

When we plug it into one of the derived equation,

With these values, we now know that

and

When we cube these values, we can see that they do indeed equal what we started with:

        

          =

          =

And more importantly, when we add the two parts together as the formula tells us to do, we get the solution, 4.

= 2 + 2

x = 4

Bombelli definitely solved Cardano’s problem, using Interstingly, neither the original problem nor the answer had anything to do with but in the method, we can see that by extending the number system to include as a valid value, it is crucial to finding the answer, as the Mathematician Jacques Hadamard quoted, “the shortest path between 2 truths in the real domain passes through the complex domain.” However, when Bombelli succeeded in finding this solution, he discarded his discovery and considered as sophistries, or “tricks” that only exist to solve problems like these. We, as thinkers of modern mathematics, know that this is not true, and there are much more sophisticated aspects to complex numbers. (BogomolnyAlexander, Remarks on the History of Complex Numbers)

How, then, are imaginary numbers valid? First of all, we need to understand exactly what limitations real numbers have. We are already familiar with the number line; it is an infinitely long line comprised of all real numbers, positive and negative. It includes all integers, all fractions and decimals, and even irrational numbers, or numbers with infinitely long decimal places, such as or . However, there is no place for on this line, and for centuries, no mathematicians could find a place for it because of one reason; i is 3-dimensional. In other words, because of the fact that i does not fit in a “real” line, all multiples of i, positive and negative, form another line, perpendicular to the line of real numbers. In the xy- coordinate plane, i forms a third axis perpendicular to both the x-axis and the y-axis.

With this comprehension, we can further define complex numbers as functioning points or vectors in the Complex Plane. A vector is defined as a quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another in a plane. This property of i opens up exponentially many possible uses of i in the 3-dimensional physical world.

The term “imaginary” make the perception of i to be some abstract, incomprehensible mathematical fallacy by many people, and it was true, until last 2 centuries. The truth is, i is as real as any other number; many people today argue that the Cartesian name of the value, the “imaginary number” is misleading, because of all of the real potentials the value actually holds. In physics alone, complex numbers are used to calculate the amount of stress on structures, resonance, for the manipulation of large matrices in modeling various figures, and is especially used extensively when dealing with electrical current, and wavelength.

In electrical engineering, values can be divided into scalar quantities and complex quantities; scalar is what real numbers are called in the scientific language. Some examples of scalar quantities include voltage produced by a power source, the resistance of any component in an electric circuit, measured in ohms (Ω), and electrical current through a wire, measured in amps. During some circuit manipulation, electrical engineers found that in alternating current circuits, voltage, current and resistance, or in physics terminology, impedance measured in AC, were not outputting scalar quantities like other DC circuits. They instead had alternating direction and amplitude (or magnitude), which as a result, had another dimension of frequency and phase shift. Engineers found that it was impossible to organize and represent all of these non-scalar values with real numbers; therefore, they turned to complex numbers, that were multi-dimensional in nature, and could express the 2-dimentional quantity of frequency and phase shift in a single complex number. However, in physics and electronics, the letter j is used in the place of i to prevent confusion, as the letter i is used to represent the value of current. Therefore, scientists would write the complex numbers in the form of . (RobertsDonna)

In electrical science, engineers are required to calculate missing values based off of given data, using specific equations such as E = I • Z, where E = voltage, I = current, and Z = impedance. For example, if the voltage in a series circuit is 45 + j10 volts and the impedance is 3 + j4 Ω, the scientist is required to be able to calculate the current by simply using the equation and inserting the values:

amps (RobertsDonna)

In contrast to some of the math problems we solved previously, the answer to these questions remain complex, which is natural, since the value still has to represent a 2-dimensional quantity of phase changes and frequency. These data are applied to anything electronic, from computers to washing machines, from someone’s smartphones to traffic lights; imaginary numbers are being used in the “real” world everywhere, which is why there are even arguments about the terminology of “imaginary” should be edited to an updated, mathematically correct term, such as “lateral numbers” for its lateral behavior in complex planes. i is truly valid.

The concept of i existed for such a short period of time, yet what it allowed us to accomplish within that time is beyond imaginable. Society saw an explosion of technological development, improved machines, and programming; all of which would have been impossible without the understanding of i in the world run by technology and electricity. However, the most crucial achievements of i is that from a number that we considered to not exist in this world, we learned more about fundamental laws of physics, the dimensions we live in, and the world, the “real” world; we need to learn from it, and appreciate it for existing.

References   

Bogomolny, Alexander. Interactive Mathematics Miscellany and Puzzles. 2015. Article. 17 September 2016.

Knaust, Helmut. “The “Cubic Formula”.” 20 5 1998. sosmath. Article. 24 September 2016.

Merino, Orlando. A Short History of Complex Numbers. Kingston, January 2006. Document.

Roberts, Donna. Does Anyone Ever Really Use Complex Numbers? 2012. Article. 25 September 2016.

Weisstein, Eric W. “Complex Number.” 4 September 2016. — from Wolfram MathWorld. Article. 19 September 2016.