Lecture 34. Complex Numbers

Origin of the complex numbers Where did the notion of complex numbers came from?Did it come from the equation

x2 + 1 = 0 (1)

as i is defined today? No. A very long time ago people had no problem accepting the factthat an equation may have no solution. When Brahmagupta (598-668) introduced a generalsolution formula

x =−b±

√b2 − 4ac

2a

for the quadratic equation ax2 + bx+ c = 0, he only recognized positive real root.

Cardan The starting point of the notion of complex number indeed came from thetheory of cubic equations. In the 16th century, cubic equations were solved by the del Ferro-Tartaglia-Cardano formula: a general cubic equation can be reduced into a special form:y3 = py + q which can be solved by

y =3

√q

2+

√(q2

)2 − (p3

)3+

3

√q

2−√(q

2

)2 − (p3

)3. (2)

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Cardan (1501-1576) was the first to introduce complex numbers a+√−b into algebra, but

had misgivings about it. In fact, Gardan kept the “complex number” out of his book ArsMagna except in one case when he dealt with the problem of dividing 10 into two partswhose product was 40. Gardan obtained the roots 5 +

√−15 and 5 −

√−15 as solution of

the equation x(10− x) = 40 and wrote:

Putting aside the mental tortures involved, multiply 5 +√−15 by 5−

√−15,

whence the product is 40.

He made a comment that dealing with√−1 “involves mental tortures and is truely sophis-

ticated” and these numbers were “ as subtle as they are useless. ” 1

Bombelli Now we know that a cubic equation with real coefficients always has a real rooty because we can consider the graph: y3− py− q > 0 when y is a large positive number andy3 − py − q < 0 when y is a large negatively number so that the graph curve must intersectthe x-axis. On the other hand, the number inside the sqaure root in (2),

(q2

)2 − (p3

)3, could

be negative. How could the formula (2), which involves a meaningless square root of negativenumber, produce a real solution in this case?

In 1569, Rafael Bombelli (1526-1572) 2 observed that the cubic equation x3 = 15x + 4does have a root, x = 4, but by the formula (2) gives

x =3

√2 +√−121 + 3

√2−√−121

=3

√(2 +

√−1)3 + 3

√(2−

√−1)3

= (2 +√−1) + (2−

√−1)

= 4

(3)

Here he demonstrated the extraordinary fact that real numbers could be generated by imag-inary numbers. From the very first time, imaginary numbers appeared in the world ofmathematics.

However, Bombelli did not really understand it. After doing this, Bombelli commented:“At first, the thing seemed to me to be based more on sophism than on truth, but I searcheduntil I found the proof.”

1J.H. Mathews and R.W. Howell, Complex Analyisis for Mathematics and Enginerring, 5th ed., Jonesand Bartlett Publishers, 2006, p.3.

2Rafael Bombelli was the last of the great sixteenth century Bolognese mathematicians, and he publisheda book Algebra” in 1572.

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Figure 34.5 Bombelli’s Algebra published in 1572.

Descartes John Napier (1550-1617), who invented logarithm, called complex numbers“nonsense.”

Rene Descartes (1596-1650), who was a pioneer to work on analytic geometry and usedequation to study geometry, called complex numbers “impossible.” In fact, the terminology“imaginary number” came from Descartes. He wrote:

Neither the true nor failse (negative) roots are always real, somethimes they areimaginary.

When he dealt with the equation z2 = az− b2, with a and b2 both positive, Descartes wrote:“For any equation one can imagine as many roots [as its degree would suggest], but in manycases no quantity exists which corresponds to what one imagines.”

Newton and Leibniz Newton (1643-1727) agreed with Descartes. He wrote: “But itid just that the Roots of Equations should be often impossible (complex), lest they shouldexhibit the cases of Problems that are impossible as if they were possible.”3

Gottfried WiIhelm Leibniz (1646-1716), who and Newton established calculus, remarkedthat imaginary numbers are lide the Holy Ghost of Christian scriptures-a sort of amphibian,midway between existence and nonexistence.4

3Morris Kline, Mathematical Thought from Ancient to Modern Times, volume 1, New York Oxford,Oxford University Press, 1972, p.254.

4http://library.thinkquest.org/22584/temh3016.htm

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Bernoulli As time passes, mathematicians gradually redefine their thinking and began tobelieve that complex numbers existed, and set out to make them understood and accepted.

Wallis tried in 1673 to give a geometric representation which failed but was quite close.

Johann Bernoulli noticed in 1702 that

dz

1 + z2=

dz

2(1 + z√−1)

+sz

2(1− z√−1)

and he could have used it to obtain

tan−1z =1

2ilog

i− zi+ z

.

Euler During the eighteenth century, imaginary numbers were used extensively in analysis,especially by Euler for it produced concrete results. These numbers not only existed, butalso obeyed the same rules of real numbers.

In 1732, Leonhard Euler (1707-1783) introduced the notation i =√−1, and visualized

complex numbers as points with rectangular coordinates. Euler used the formula

x+ iy = r(cos θ + isin θ), r =√x2 + y2

and visualized the roots of zn = 1 as vertices of a regular polygon, which was used beforeby Cotes (1714) 5. Euler defined the complex exponential, and proved the famous identity

eiθ = cos θ + i sin θ.

Although complex numbers were being admitted, doubt concerning their precise meaningcontinued to puzzle mathematicians. In fact, even Euler had made a remark:

5John Stillwell, Mathematics and its history, Second edition, Springer, 2002, p.263.

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“such numbers, which by their natures are impossible, are ordinarily called imag-inary or fanciful numbers, because they exist only in the imagination.

Wallis Three centuries passed after their introduction, in absence of a full and convincingunderstanding of these new numbers, progress in their foundation was developed, either ingeometric way, or in arithmetic way.

John Wallis might be the first to attempt, although unsuccessfully, a graphic represen-tation of a complex number. In his book Algebra in 1685, he suggested to use Euclideangeometry to deal with complex numbers.

Wessel and Argand Caspar Wessel (1745-1818) first gave the geometrical interpretationof complex numbers

z = x+ iy = r(cos θ + isin θ)

where r = |z| and θ ∈ R is the polar angle. Wessel’s approach used what we today callvectors. He uses the geometric addition of vectors (parallelogram law) and defined multi-plication of vectors in terms of what we call today adding the polar angles and multiplyingthe magnitudes. Wessel’s paper, written in Danish in 1797.

The same fate awaited the similar geometric interpretation of complex numbers put forthby the Swiss bookkeeper J. Argand (1768-1822) in a small book published in 1806.

Gauss It was only because Gauss used the same geometric interpretation of complexnumbers in his proofs of the fundamental theorem of algebra and in his study of quarticresidues that this interpretation gained acceptance in the mathematical community. 6

The realization that the use of complex numbers enabled a polynomial of degree n tohave n roots might lead to their reluctant acceptance. In 1799 Carl Friedrich Gauss gave the

6Victor J. Katz, A History of Mathematics - an introduction, 3rd edition, Addison -Wesley, 2009, p.796.

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first of his four proofs for the well-known Fundamental Theorem of Algebra: Any polynomialequation

anxn + an−1x

n−1 + ...+ a1x+ a0 = 0 (an 6= 0)

has exactly n complex roots. Although the name fundamental theorem of algebra and proofswere given by Gauss, the result itself was known to mathematicians such as d’Alembert(1746), Euler (1749) and Language (1772).

In 1811 Gauss wrote to Bessel to indicate that many properties of the classical func-tions are only fully understood when complex arguments are allowed. In this letter, Gaussdescribed the Cauchy integral theorem 7but this result was unpublished.

Gauss

Hamilton Rowan Hamilton (1805-65) in an 1831 memoir defined ordered pairs of realnumbers (a, b) to be a couple. He defined addition and multiplication of couples: (a, b) +(c, d) = (a+ c, b+d) and (a, b)(c, d) = (ac− bd, bc+ad). This is the first algebraic definitionof complex numbers.

7For Cauchy, see the next chapter.

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