"Maxwell's equations are differential equations for the electric (E) and magnetic (H) field vectors because these fields, at every point in space, for every instant of time, can be related to the fields at nearby points in space and time. They are partial differential equations because there are multiple independent variables, i.e., time, and at least one space variable." (Paul J Nahin, "The Science of Radio", 1995)
"What Maxwell expressed mathematically in his famous set of partial differential equations is (1) electric lines of force are created either by electric charge or by time-varying magnetic fields, and (2) magnetic lines of force are created either by currents (moving electric charge) or by time-varying electric fields. The last half of (2) is uniquely Maxwell, as it represents his famous displacement current. That a time-varying electric field in space could produce a magnetic field, just like a conduction current in a wire, was an audacious statement by Maxwell because at that time there simply was not the slightest experimental evidence for it." (Paul J Nahin, "The Science of Radio", 1995)
"While amplitude damped spark gap radio waves are suitable for transmitting keyed on-off coded signals, such waves are not suitable for voice (or music) transmission. The reason for this can be understood once it is realized that for information to be transmitted it is necessary for something to change. If nothing changes, then the receiver of the signals can predict with perfect accuracy what the future nature of the signals will be - exactly what they werel It is unnecessary, actually, to transmit anything if nothing changes." (Paul J Nahin, "The Science of Radio", 1995)
"How beautifully simple is Wessel’s idea. Multiplying by √-1 is, geometrically, simply a rotation by 90 degrees in the counter clockwise sense [...] Because of this property √-1 is often said to be the rotation operator, in addition to being an imaginary number. As one historian of mathematics has observed, the elegance and sheer wonderful simplicity of this interpretation suggests 'that there is no occasion for anyone to muddle himself into a state of mystic wonderment over the grossly misnamed ‘imaginaries'. This is not to say, however, that this geometric interpretation wasn’t a huge leap forward in human understanding. Indeed, it is only the start of a tidal wave of elegant calculations." (Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998)
"The discovery of complex numbers was the last in a sequence of discoveries that gradually filled in the set of all numbers, starting with the positive integers (finger counting) and then expanding to include the positive rationals and irrational reals, negatives, and then finally the complex." (Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998)
"When we try to take the square root of -1 (a real number), for example, we suddenly leave the real numbers, and so the reals are not complete with respect to the square root operation. We don’t have to be concerned that something like that will happen with the complex numbers, however, and we won’t have to invent even more exotic numbers (the ‘really complex’!) Complex numbers are everything there is in the two-dimensional plane." (Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998)
"[…] and unlike the physics or chemistry or engineering of today, which will almost surely appear archaic to technicians of the far future, Euler’s formula will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)
"Being able to appreciate beautiful mathematics is a privilege, and many otherwise educated people who can't sadly understand that they are 'missing out' on something precious." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)
"I think e^iπ+1=0 is beautiful because it is true even in the face of enormous potential constraint. The equality is precise; the left-hand side is not 'almost' or 'pretty near' or 'just about' zero, but exactly zero. That five numbers, each with vastly different origins, and each with roles in mathematics that cannot be exaggerated, should be connected by such a simple relationship, is just stunning. It is beautiful. And unlike the physics or chemistry or engineering of today, which will almost surely appear archaic to technicians of the far future, Euler's formula will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)
"[...] one of the fundamental intellectual breakthroughs in the historical understanding of just what i = √-1 means, physically, came with the insight that multiplication by a complex number is associated with a rotation in the complex plane. That is, multiplying the vector of a complex number by the complex exponential e^iθ rotates that vector counterclockwise through angle θ." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)
"Ugly creations, in my opinion, be they theories or paintings, are ones that obey no constraints, that have no discipline in their nature." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)
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