The Story of Complex Numbers

Complex Numbers are numbers of the form a + bi, where a and b are Real Numbers and i = √-1. a is called the real term and bi the imaginary. Indeed, no one before the 18th century could imagine a number such as i. The story of Complex Numbers has to start with that of Negative Numbers. The latter are familiar to us now, but before the 17th century, there was no clear notion of negative numbers. The story goes that Blaise Pascal (1623-1662) once remarked "Who doesn't know if you take 4 away from nothing, you're still left with nothing?" However, before Pascal, there were already people working on negative numbers. Brahmugputa (600-670) was working regularly with negative numbers. For the western community, Thomas Harriot (1560-1621) is amongst the first and Raphael Bombelli (1526-1573) even gave clear definitions. The following statements are due to Bombelli:

(+)(+) = +
(+)(-) = -
(-)(+) = -
(-)(-) = +

Simon Stevin (1548-1620) and Albert Girard (1595-1632) treated negative numbers as being on par with positive ones.

In 1545, Girolamo Cardano (1501-1576) solved the equation x(10-x) = 40 in his famous book Ars Magna giving the solutions 5 � √-15. In the same book he gave the solution to the 'Depressed Cubic' x3 = 15x + 4 as

x = 3√[2 + √-121] + 3√[2 - √-121].

Since the obvious solution is x = 4, something was clearly amissed. Bombelli observed that

(2 � √-1)3 = 2 � √-121.

So Cardano's solution becomes

x = (2 + √-1) + (2 - √-1) = 4

Bombelli wrote the following in his work Algebra (but he did not used the symbol i):

(+i)(+i) = -1
(+i)(-i) = +1
(-i)(+i) = +1
(-i)(-i) = -1

In 1637, Ren� Descartes (1596-1650) called the expressions involving square root of negative numbers 'imaginary' and took their occurrence as a proof that the problem is unsolvable. Isaac Newton (1642-1727) shared his view. Then, Johann Bernoulli (1667-1748) encountered Logarithms of Complex Numbers while trying to integrate functions like [x2 + 3x + 1]-1. By 1712, Bernoulli and Gottfried William Leibniz (1646-1716) were engaged in a battle over such Logarithms. Bernoulli argued that since d(-x)/(-x) = dx/x, then integration shows that ln(-x) = lnx. Leibniz however, maintained that integration works only for positive x. Leonhard Euler (1707-1783) said they were both wrong, that integration requires an arbitrary constant, so that the correct conclusion is ln(-x) = lnx + c, for some constant c, which obviously equal to ln(-1). Euler then carry on to prove the strange but interesting formula

eix = cosx + isinx

Actually, Roger Cotes (1682-1716) has found much the same result in 1714. Setting x = π, we obtain the extraordinarily beautiful formula (also called Euler's Formula)
eiπ = -1,

which links the 4 fundamental constants in Mathematics together. In logarithmic form this equation becomes ln(-1) = iπ. However, by cubing Euler's Formula and taking logarithm, we get ln(-1) = 3iπ. In fact, we can take any odd power of Euler's Formula to obtain multiple values for ln(-1). If L is one of them, then the others are L � 2πi, L � 4πi, ...


John Wallis (1616-1703), in his Treatise on Algebra (1673), interpreted a complex number as a point in the plane, but it was ignored. In 1797, the Norwegian surveyor Caspar Wessel (1745-1818) published a Danish paper showing how to represent complex numbers in a plane. This also went unnoticed, until it was translated into French a century later. Meanwhile, the idea became attributed to Jean Robert Argand (1768-1822), who wrote it independently in 1806. Hence the complex plane is also known as an Argand Diagram.

By 1811, Carl Friedrich Gauss (1777-1855) was habitually thinking of complex numbers as points in a plane and he published his ideas in 1831. Finally, in 1837, William Rowan Hamilton (1805-1865) took the last step by identifying a + bi with its coordinates (a,b) and rewriting the geometric definitions in algebraic form. Actually, Gauss had already anticipated Hamilton's algebraic formulation in 1832. As it turned out, whenever someone made a discovery in the mid-1800s, Gauss seemed to have already thought of it, but did not reveal to anyone.

Some last words are in place. It seems that the field of Complex Numbers is really very rich and interesting, for most of the greatest mathematicians from the mid-1500s to the mid-1800s had a hand in it. In fact, there has been pretty much developments even until recently, with the advent of Complex Analysis and applications to the physical world, in particular, Electrical Engineering.

Short notes on some of the mathematicians mentioned in this page.
1. Thomas Harriot (English) made contributions to Optics (e.g. he discovered Snell's Law of Refraction before Snell), Astronomy (e.g. first to discover sunspot and a moon of Jupiter), Algebra, the Sphere-packing Problem. Despite all these discoveries, he published nothing in his life. His greatest work Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas was published in 1631, 10 years after his death.
2. Raphael Bombelli's (Italian) work Algebra consisted of 5 books (Book I to III are algebraic and Book IV to V are geometric). Some of the problems in the books are due to Diophantus, some due to himself; and he developed on Cardano's ideas, formalizing the concept of negative numbers and complex numbers. His work differ from Cardano's in that he made extensive use of symbols, whereas Cardano did not used any at all.
3. Simon Stevin (Dutch) contributed to Algebra, Trigonometry, Mechanics, Hydrostatics. He was an outstanding engineer who built windmills, locks and ports. His De Thiende (The Tenth) (1585) presented a thorough account of decimal fractions. He is the first to used the symbols +, -, √.
4. Albert Girard (French) worked on Arithmetic, Algebra and Trigonometry. His 1626 work on Trigonometry is the first to use the symbols sin, cos, tan. He is the first to give a recursive definition to the Fibonacci Sequence.
5. Ren� Descartes' (French) most famous work La G�om�trie made him the discoverer of the Cartesian system. He also worked on Optics and Meteorology. His work on the latter, although highly flawed, is the first scientific attempt to study the weather. His Vortex Theory of the Universe advocated that the universe is filled with matter, which we now know is false.
6. Gottfried William Leibniz (German) is famous for his rivalry with Newton as the contender for founding Calculus. In fact, he was the one to introduce the notations dy/dx and ∫f(x)dx.
7. Roger Cotes (English) contributed to Logarithms, Integral Calculus and Numerical Methods. He only published one paper in his life, namely Logometria, which is a work on Logarithm. After his death, some of his works were published in the 2 volumes of The Doctrine and Application of Fluxions.
8. John Wallis is considered the most influential English mathematician before Newton. Among his works are Arithmetica infinitorum (1656) and Tract on Conic Sections (1656). He introduced the symbol ∞ for infinity.
9. Caspar Wessel (Norwegian) published only one single paper (mentioned above). He is also known for his triangulation of Denmark.
10. Jean Robert Argand (Swiss) was an amateur mathematician. He interpreted i as a rotation through 90o and defined the modulus of a complex number. He provided a slightly-flawed but beautiful proof of the Fundamental Theorem of Algebra in 1814.
11. William Rowan Hamilton (Irish) is the discoverer of the Quaternion; and he used the characteristic function for optics and dynamics to great success.

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