Abstract
In this introductory chapter, we give a brief introduction of the complex number system, geometrical representation of complex numbers, the notion of point at infinity, Riemann sphere, and metric topology of \(\mathbb {C}\). All these notions are meant to convey the need for and the intrinsic beauty found in passing from a real variable x to a complex variable z.
If I were again beginning my studies, I would follow the advice of Plato and start with mathematics
Galileo Galilei
Science is what we understand well enough to explain to a computer, Art is all the rest
Donald E. Knuth
I used to love mathematics for its own sake, and I still do, because it allows for no hypocrisy and no vagueness
Stendhal (Henri Beyle), The Life of Henri Brulard
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Notes
- 1.
The term “complex number” first occurs in 1831, C. F. Gauss, Werke, II. p. 102.
- 2.
First to use complex numbers for this purpose was the Italian mathematician, G. Cardano (1501–1576), who found the formula for solving cubic equation.
- 3.
The use of i instead of \(\sqrt{-1}\) is due to L. Euler in 1748.
- 4.
W. R. Hamilton’s quaternions furnish an example of a still further extension of the idea of complex number.
- 5.
K. Weierstrass calls the modulus of \(x+iy\) as the absolute value of \(x+iy\) and writes it as \(|x+iy|\).
- 6.
Named after the French mathematician Jean Robert Argand(1768–1822), born in Geneva and later a librarian in Paris. His paper on the complex plane appeared in 1806; it had however previously been used by Gauss, in his Helmstedt dissertation, 1790 (Werke, III. pp. 20–23), who had discovered it in October 1797 (Math. Ann. LVII. p. 18), and C. Wessel had discussed it in a memoir presented to the Danish Academy in 1797 and published by that society in 1798–99.
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Pathak, H.K. (2019). Complex Numbers and Metric Topology of \(\mathbb {C}\). In: Complex Analysis and Applications. Springer, Singapore. https://doi.org/10.1007/978-981-13-9734-9_1
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DOI: https://doi.org/10.1007/978-981-13-9734-9_1
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