The Evolution of Complex Number

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The Evolution of Complex Numbers

Leigh Walton

i

Practical Applications of Advanced Mathematics

Amy Bowman


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Abstract

Complex numbers have had a long and twisted history, largely because medievalmathematicians were consistently uncomfortable working with these concepts, which they

considered impossible or useless. This sentiment is still popularly echoed today, even

by those who are relatively comfortable with mathematics.

Nevertheless, imaginary and complex numbers have finally taken their rightful place as

full members of our system of numeration, thanks to significant gains made in the last 300years. Initially developed to handle previously unsolvable quadratic and cubic

equations, the idea of complex numbers has grown tremendously, to integrate with

coordinate systems, vectors, matrices, and even quantum mechanics. The concept of

numbers placed together in pairs or sets, along with a convenient system of manipulatingthem, has such widespread use that Cardano, Descartes, or Euler would be awestruck to

see the fruits of their ideas. As we discover more about quantum mechanics and other

advanced physics, complex numbers continue to grow ever more significant.

Background (What is a Complex Number?)

Essentially, all numbers are complex numbers. The problem arises from taking the real

numbers, which most people are comfortable with, and moving outside them.

The whole numbers are {0,1,2,3, . . .}. The integers are {. . . 2, -1, 0, 1, 2, . . .}. The

rational numbers are any any number that can be expressed as a fraction of two integers,

such as 4 or or . Irrational numbers are all the real numbers that are not rational

numbers. The real numbers are all the numbersx such thatx < 0,x = 0, orx > 0. Purely


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imaginary numbers are all the numbers bi such that b is a real number and i is

( 1 ) . In other words, i times i is 1. The complex numbers, then, are any

sum a + bi, where a and b are real numbers. Notice that either or both ofa and b could be

0.

An essential part of complex numbers is their dual nature: each has a real part (a)

and an imaginary part (b). This duality is very significant to later applications.

History

The first recorded appearance of the square root of a negative number is in the

Stereometria of the 1st-century mathematician Heron of Alexandria. In it, Heron attempted

to solve a quadratic equation for the side of a pyramid and ran into ( 8 1 1 4 4 ). Naturally,

the correct answer is ( 6 3 ), but for some reason Heron wrote ( 6 3 ) , through either

his own error or a copyists.

Around A.D. 250, Diophantus of Alexandria wrote another quadratic equation that

would have led to a complex solution ( , to be precise), had he finished the

calculation. However, he stopped in the middle and concluded that his problem was

impossible. Interestingly, he did so not because it contained the root of a negative number,

but simply because it contained a negative number at all!

During the sixteenth century, mathematicians attempted to solve various cubic

equations, or equations in which the variable is raised to the third power and no higher.

Scipione del Ferro found a formula for the cubicx3 + px = q sometime around 1600, but

kept it a secret because the mathematical climate of those days was one of competition,

contests, secrecy, and fame. When someone discovered a new technique, he would

challenge another to a problem-solving contest and hopefully win glory and the favor of a


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rich patron. Still, del Ferro told almost no one except his student, Antonio Fior, who

promptly challenged a far more talented mathematician, Tartaglia. Tartaglia, fearing that

Fior would use del Ferros secret, attempted to solve the cubic himself and finally did it

before the contest, easily besting Fior.

Tartaglia, too, valued his secrecy, and refused to reveal his techniques until

Girolamo Cardano begged him for the answer. Once Cardano realized that Tartaglia was

not the first to solve the cubic, he felt free to publish the solution in his bookArs Magna,

giving del Ferro and Tartaglia credit. In fact, he generated a formula applicable to all

cubic equations, as well as performing the first actual calculations with complex numbers.

He invented the problem of splitting 10 into two parts whose product is 40, or (x)(10 x) =

40. This equation has the solutions 5 + and 5 - , whose impossible nature

Cardano temporarily ignored while multiplying them. Indeed, he got 40, but after all this

he concluded that all his work was useless.

In 1572 Rafael Bombelli noticed that some cubics with obviously real solutions

generated bizarre complex numbers when put through the Cardano cubic formula. He

proposed that these complex monstrosities (such as ) were actually

the obviously real solutions (such as 4), and went through some complicated algebra to

prove it.

It was a common practice in the days of Ren Descartes to represent geometric

problems with algebra and vice versa. What Descartes knew was that when these algebraic

formulae popped up with imaginary numbers, the geometry failed. The line segment

didnt reach the expected point; the triangle was not formed; the angle didnt add up right.

No wonder he and his contemporaries still considered imaginary numbers to be


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impossible. Of course, the same thing happens sometimes when the algebra generates a

negative number (i.e. for a side length).

The first person to get close to a graphical representation of complex numbers was

the Englishman John Wallis. He never achieved an actual system of representation, but his

work on a geometric/algebraic problem hints toward the system we use today. Essentially,

he worked with the SSA triangle, in which two side lengths are known and an angle which

is not between them. If one of the sides is too short (i.e. if the algebra produces an

imaginary number), then the triangle is not formed. Wallis, however, realized that a

triangle is formed, but only if one shifts the base upward. The resulting triangle does not

contain the original angle, but it does hint toward our understanding: imaginary numbers

indicate a vertical movement in the plane. This idea is established much more clearly

later.

Still, advances were made using a purely algebraic concept of ( 1 ) , most

notably by the incredible Leonhard Euler, the most prolific writer of mathematics ever.

Using calculus, he proved that the numbere, when raised to the imaginary powerbi, is

equal to the cosine ofb plus i times the sine ofb. Mathematically, ebi = cos b + i sin b.

This equation has tremendous significance, allowing huge numbers of theorems to be

constructed and stepping closer to the unification of mathematics. Euler also discovered

that ii is actually a real number, and has many different values simultaneously!

Still, many refused to dabble with such impossible numbers until they saw a

geometric representaion. In fact, nearly the whole of our complex number geometry was

established all at once by a Norwegian surveyor, Caspar Wessel, in 1797. His paper On

the Analytic Representation of Direction: An Attempt, the first paper submitted to the

Danish Academy of Sciences by a non-member, established a perfectly sensible system for


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handling complex numbers. He based his system on what we now call vector addition,

the addition of directed line segments. Each positive number can be represented by an

arrow pointing right on a number line, of length equal to the number. Similarly, each

negative number is an arrow pointing to the left. Hence from zero, a positive number + its

opposite yields zero again, and 2 + 5 yields +3. Wessel (and Wallis, actually) reasoned

that non-horizontal arrows should obey the same rule for addition: following each arrow

and then starting the next from that point.

Wessels contribution was inventing a representation ofmultiplying these directed

segments. Each segment has two properties: length and direction. The latter can be

measured by an angle that it forms measuring counterclockwise from the x-axis. Hence,

when multiplying two directed segmentsy andz, Wessel suggested the product will have

length equal to the product of the lengths ofy and z, and an angle equal to the sum of the

angles ofy andz. This isnt quite connected with imaginary numbers yet, but well soon

see that it is essential.

Every real number can be represented by a point on a number line. Numbers with

an imaginary part cannot be placed on a number line, so they must go outside it. If we

make this number line the x-axis, and use Wessels rules for multiplication, we can easily

see how complex numbers fit in. Ifi really can be represented by a point or a vector, then

multiplying it by itself must equal 1. Giving i a length of r and an angle of , then (r, )

times (r, ) = 1. Wessels rule for multiplication tells us that (r, ) (r, ) = (r2,2 ). 1


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can be seen as a point as well: its length is 1 and it forms a 180angle to the x-axis. 1 is

the point (1,180). If (r2,2 ) = (1,180), then r2 = 1 and 2 = 180. Hence, r is 1 and is

90, which places i straight up, one unit above the origin.

This conclusion, which seems so simple now, was and is astonishingly significant.

At last, complex numbers were understandable, for its clear that 3 +2 i must lie three units

to the right and two units up. Cardanos 5 + and 5 - , which perplexed him

earlier, are simply two points: one five units right and units up, and the other five

units right and units down. If we found the lengths and angles of each point, we could

use Wessels multiplication to verify that they do multiply to the point 40.

Sadly, Wessels paper outlining all these ideas was written only in Danish, and

published only in a small journal read mostly inside Denmark. Indeed, his paper was

virtually ignored until 1895, when it was rediscovered and recognized. Most of the credit

for the geometric interpretation, therefore, commonly goes to the Swiss bookkeeper Jean-

Robert Argand. He published a small paper in 1806, essentially proposing the same ideas

as Wessel, without even placing his own name on the title page. This, too, would have

disappeared, if one of the men who received the paper had not mentioned it to a friend,

who died and whose brother published the ideas in a well-known journal and invited

Argand to claim the ideas. Ever since, the plane of real and imaginary axes has been

known as the Argand diagram.

After Argand, the ideas of complex numbers were developed further, notably by

the William Rowan Hamilton and Carl Friedrich Gauss in the first half of the 19th century.

Hamilton disliked the geometric representation and devised a system based purely on

number couples, or ordered pairs. He developed formal rules for developing these


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couples algebraically, which when viewed from the geometric perspective, really turn

out to be the same rules developed earlier for Wessels complex numbers.

Gauss, an intellectual giant, had apparently been working with complex numbers

for some time when all these works were published, and approved of the results. His

reputation was very influential in gaining acceptance for ( 1 ) . In his honor,

the plane is also sometimes called the Gaussian plane.

Much, much more has been done with imaginary numbers since Gauss, involving

calculus, 3-dimensional systems, and other higher maths, but these beginnings are

sufficient to illustrate the convoluted history ofi.

Attitudes

The history of complex numbers is perhaps most intriguing because, unlike the

discovery of many other concepts, that of complex numbers is surrounded by controversy,

unease, and secrecy. Indeed, the evolution ofi has been influenced as much by attitudes

toward it as by actual mathematical discovery.

Many early mathematicians shared the opinion of the 9 th-century Hindu

Mahaviracarya, The square of a positive number, as that of a negative number, is positive.

Hence the square root of a positive number is twofold, positive and negative. There is no

square root of a negative number, for a negative number is not a square. After putting

aside the mental tortures involved and performing simple operations on ( 1 5 ) , the

sixteenth-century Italian Girolamo Cardano decided his method was as refined as it is

useless. When the early mathematicians ran intox2 + 1 = 0 and other such quadratics

they simply shut their eyes and called them impossible, according to Paul Nahin (1998).


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Centuries later, while mathematicians accepted that ( 1 ) could produce real

results (using the idea of conjugates), they were still uncomfortable in its use. Still

grounded heavily in the Greek tradition of geometry, they disliked any number lacking a

graphical representation. One can draw a line of length 1, or 5, or 2 , but not of length

( 1 ) . Neither can one place 2+ ( 3 ) on a number line. Hence we have

Ren Descartes in 1637, working from polynomial equations, coining the term

imaginary: Neither the true nor the false roots are always real; sometimes they are

imaginary; that is, while we can always conceive of as many roots for each equation as I

have already assigned, yet there is not always a definite quantity corresponding to each

root so conceived of.

Leonhard Euler, despite his tremendous work with complex numbers, in 1770

wrote:

All such expressions as ( 1 ) , ( 2 ) , etc., are

consequently impossible or imaginary numbers, since they represent roots

of negative quantities; and of such numbers we may truly assert that they

are neither nothing, nor greater than nothing, nor less than nothing, which

necessarily constitutes them imaginary or impossible.

(Nahin, 1998, p.31)

Even after the work of Wessel and Argand developing a geometric interpretation,

there was still resistance. Their contemporary, Francois-Joseph Servois wrote, I confess I

do not yet see in this notation anything but a geometric mask applied to analytic forms the

direct use of which seems to me simple and more expeditious. Hamiltons friend

Augustus De Morgan wrote, we have shown the symbol ( 1 ) to be void of

meaning, or rather self-contradictory or absurd. As late as 1854, the logician George


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Boole called i an uninterpretable symbol, and the same unfortunate opinion is held by

millions today. Luckily, most mathematicians, engineers, and physicists have rid

themselves of this bias and discovered myriad applications for complex numbers.

Applications

Complex numbers is such a broad subject that it naturally has many applications.

Many of them are centuries old, while some have only been discovered recently.

One of the most clever and accessible is an example from George Gamows book

One, Two, Three . . . Infinity, in which a man attempts to find a buried treasure using the

instructions contained in a secret message (pp. 35-8). Unfortunately, one of the landmarks

mentioned in the message has disappeared, but some complex number geometry makes it

simple to find the treasure, wherever the landmark may have been.

In An Imaginary Tale, Paul J. Nahin describes several electrical circuits whose

operation can only be measured and evaluated using imaginary numbers(pp.125-141). At

the International Electrical Conference of 1893, Charles Steinmetz wrote:

We are coming more and more to use these complex quantities instead of

using sines and cosines, and we find great advantage in their use for

calculating all problems of alternating currents [AC], and throughout the

whole range of physics. Anything that is done in this line is of great

advantage to science.

(Nahin, 1998)

Complex numbers have many applications in pure mathematics. They prove the

Fundamental Theorem of Algebra (that each polynomial equation has a number of


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solutions equal to its degree), they find complex roots of any number, they derive

trigonometric identities, and they are the foundation of a whole branch of mathematics:

complex function analysis.

Physicists also find complex numbers useful. Combined with vector mathematics,

complex numbers can model forces and motion. They also are extremely useful for

interpreting and predicting the motion of heavenly bodies, and for analyzing the flow of

fluids such as air or water around a moving object, such as an airplane wing. The field of

quantum mechanics make extensive use of probability amplitudes, which are essentially

complex numbers.

Conclusion

John Stillwell sums it up rather nicely:

At the beginning of their history, complex numbers a + b ( 1 ) were

considered to be impossible numbers, tolerated only in a limited algebraic

domain because they seemed useful in the solution of cubic equations. But

their significance turned out to be geometric and ultimately led to the

unification of algebraic functions with conformal mapping, potential theory,

and another impossible field, non-Euclidean geometry. This resolution of

the paradox ( 1 ) was so powerful, unexpected, and beautiful that

only the word miracle seems adequate to describe it.

(Nahin, 1998)

This is the true beauty of complex numbers: the astonishing vastness of the field, the

incredibly wide applications of the concept, the elegant simplicity of it all. Theres a

beauty, too, in the irony of a concept so commonly reviled and misunderstood that is so


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pervasive and simple, and in a history filled with mistrust and failed attempts, the

breathtaking simplicity of the method of a surveyor from Norway.


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References

Biggus, Jeff. (2000). Sketching the history of hypercomplex numbers. RetrievedJuly 13, 2001, from http://history.hyperjeff.net/hypercomplex.html.

Budden, F. J. (1970). Complex numbers and their applications. London: Longman

Group. (Original work published 1968).

Gamow, G. (1988). One, two, three--infinity. New York: Dover Publications.

(Original work published 1947).

Nahin, P. J. (1998). An imaginary tale : The story of [the square root of minus one].

Princeton, NJ: Princeton University Press.

Spreckelmeyer, R. (1965). The complex numbers. Boston: D. C. Heath and

Company.

Stewart, I. & Tall, D. (1983). Complex analysis : The hitchhiker's guide to theplane. Cambridge: Cambridge University Press.

Descartes, Ren. (1952). La gomtrie. (D. E. Smith and M. L. Latham, trans.)

Great books of the western world. Chicago: Encyclopdia Britannica. (Original work

published 1637).
http://history.hyperjeff.net/hypercomplex.htmlhttp://history.hyperjeff.net/hypercomplex.html

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The Evolution of Complex Number