On Complex Numbers V

“It is generally true, that wherever an imaginary expression occurs the same results will follow from the application of these expressions in any process as would have followed had the proposed problem been possible and its solution real.” (Augustus de Morgan, “On the Study and Difficulties of Mathematics”, 1898) 

“What could be more beautiful than a deep, satisfying relation between whole numbers. How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex numbers.” (Manfred Schroeder, “Number Theory in Science and Communication”, 1984)

“The attitudes of mathematicians can be found not only in what they wrote, but in what they did not write. It is possible to divide mathematicians into those who gave complex numbers some kind of coverage, and those who sometimes or always ignored them.” (Diana Willment, “Complex Numbers from 1600 to 1840” [Masters thesis], 1985)

“The lack of a visual representation for √-1 had a profound influence on attitudes to it, and complex numbers were not widely accented until after the invention of the Argand diagram.” (Diana Willment, “Complex Numbers from 1600 to 1840” [Masters thesis], 1985)

“The square roots of negative numbers! If negative numbers were false, absurd or fictitious, it is hardly to be wondered at that their square roots were described as 'imaginary'.” (David Wells, “The Penguin Dictionary of Curious and Interesting Numbers”, 1986)

“The original purpose and immediate objective in introducing complex numbers into mathematics is to express laws of dependence between variables by simpler operations on the quantities involved. If one applies these laws of dependence in an extended context, by giving the variables to which they relate complex values, there emerges a regularity and harmony which would otherwise have remained concealed.” (Heinz-Dieter Ebbinghaus et al, “Numbers”, 1990)

"The number ‘i’ is evidence that much real progress can result from the positing of imaginary entities. Theologians who have built elaborate systems on much flimsier analogies should perhaps take heart." (John A Paulos, “Beyond Numeracy”, 1991)

“If we think of square roots in the geometric manner, as we have just done, to ask for the square root of a negative quantity is like asking: ‘What is the length of the side of a square whose area is less than zero?’ This has more the ring of a Zen koan than of a question amenable to a quantitative answer.” (Barry Mazur, “Imagining Numbers”, 2003)

“To have the courage to think outside the square, we need to be intrigued by a problem. This intrigue will encourage us to use our imaginations to find solutions which are beyond our current view of the world. This was the challenge that faced mathematicians as they searched for a solution to the problem of finding meaning for the square root of a negative number, in particular √-1.” (Les Evans, “Complex Numbers and Vectors”, 2006)

“Unfortunately, if we were to use geometry to explore the concept of the square root of a negative number, we would be setting a boundary to our imagination that would be difficult to cross. To represent -1 using geometry would require us to draw a square with each side length being less than zero. To be asked to draw a square with side length less than zero sounds similar to the Zen Buddhists asking ‘What is the sound of one hand clapping?’” (Les Evans, “Complex Numbers and Vectors”, 2006)

See also:
5 Books 10 Quotes: Complex Numbers V
Complex Numbers III
Complex Numbers II
Complex Numbers I