Download 1 A Brief History of √−1 and Complex Analysis

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A Brief History of
√
−1 and Complex Analysis
• The quadratic formula has been known since Babylonian times, but negative square roots
were seen as coming from impossible solutions and were discarded.
• The depressed cubic equation (2nd order term removed) was first solved by Scipione del
Ferro (1465–1526) who kept it secret. (University positions could be won and lost through
mathematical challenges.) His solution to x3 + px = q (both p and q positive) is equivalent
to
�
�
�
�
3
2
3 q
3 q
p
p3
q
q2
+
+
+
−
+
x=
2
4
27
2
4
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Cubics were all reduced to one of three cases without
negative coefficients: x3 + px = q,
√
x3 = px + q, and x3 + q = px. Everyone still viewed −1 as an impossible quantity.
• The depressed cubic was rediscovered by Tartaglia (Niccolo Fontana 1500–1577) in 1535.
• Girolamo Cardano (1571–1576) learned the formula from Tartaglia under an oath of secrecy.
Later, Cardan was shown a book written by del Ferro and saw that as releasing his oath.
He published a method in Ars Magna giving credit to both del Ferro and Tartaglia. Cardan
extended the solution to all cubics x3 +a1 x2 +a2 x+a3 = 0 via the transformation x = X−a1 /3.
He called equations with complex roots sophistic, but used them formally to obtain real roots
using the identity z + z̄ = 2Re(z).
• Rafael Bombelli
(1526–1572) wrote of a “wild idea” in his Algebra of 1542: Formal manipu√
lation of −1 to obtain real results.
• François Viète (1540–1603) used cosine and arccosine to avoid imaginary quantities.
• Leonhard Euler (1707–1783) wrote to Johann Bernoulli (his former teacher) in 1740 the result
now known as “Euler’s Formula”
√
√
eθ −1 = cos(θ) + sin(θ) −1
He published the formula in 1748 in his Introductio Analysis Infinitorum. The result had been
published first by Roger Cotes (1682–1716) in 1714, but in logarithmic form. Cotes writing
was so obscure that no one understood his work.
• Abraham De Moivre (1667–1754) never published “his” theorem
√ �1/m
√
�
= cos (θ/m) + sin (θ/m) −1
cos(θ) + sin(θ) −1
but, Isaac Newton and Leonhard Euler (1748) attributed it to him
• Jean d’Alembert (1717–1783) presented a proof of the Fundamental Theorem of Algebra using
complex values. His proof was flawed, but led Euler, Gauss, and others to work on the result.
• Euler invented the ι notation in 1777. Euler wrote in his Algebra in 1770
√
√
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.
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Euler also gave a “proof” that
√
−2 ·
√
−3 =
√
6.
• Caspar Wessel (1745–1818) invented the geometric interpretation as x + ιy in xy-plane in
1797. His work had no impact outside Denmark until his paper was rediscovered in 1895.
Wessel presented complex multiplication geometrically as the product of magnitudes and sum
of arguments: To multiply by ι is a counter-clockwise rotation by 90 degrees.
• Jean-Robert Argand (1768–1822) rediscovered Wessel’s idea independently in 1806.
• Adrien-Quentin Buée (1748–1826) also rediscovered Wessel’s idea in 1806. He stated
√
“ −1 is the sign of perpendicularity.”
• Augustin-Louis Cauchy (1789–1857) investigated functions of a complex variable. He developed the calculus of complex functions (1814), discovered the connection between derivative
and integral, investigated infinite series, and defined analytic functions.
• Karl Freidrich Gauss (1777–1855) introduced the term complex number. He proved the Fundamental Theorem of Algebra and in 1849 extended the result to complex polynomials.
• William Rowan Hamilton (1806–1865) defined complex numbers as ordered pairs such that
(a, b) + (c, d) = (a + b, c + d)
and
(a, b)(c, d) = (ac − bd, ad + bc)
in 1835. Hamilton wanted a purely algebraic definition of complex numbers.
• Karl von Weierstrass (1815–1897) proved (1863) that the complex numbers are the only
commutative algebraic extension of the real numbers. Gauss had promised, but failed, to
give a proof of this theorem.
• Victor Puiseux (1820–1883) defined branches and was the first to distinguish poles, essential
points and branch points.
• Georg Fredreich Bernhard Riemann’s (1826–1866) doctoral thesis (1851), directed by Gauss,
studied the theory of complex variables and what we now call Riemann surfaces, introducing
topological methods into complex function theory.
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People of Complex Analysis
Scipione del Ferro
Nicollo Tartaglia
Gerolamo Cardan
François Viète
Leonhard Euler
Johann Bernoulli
Roger Cotes
Auguste De Moivre
Issac Newton
A. D’Alembert
Augustin Cauchy
Karl Gauss
William Hamilton
Karl Weiertrass
Victor Puiseux
Georg Riemann
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Early Texts with
√
−1
The images below show the frontispieces of the texts.
Girolamo Cardano
Rafael Bombelli
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