A bunch of us engineers were sitting together for lunch in a company cubicle when we were interrupted by someone whom the company had hired as a resident mathematician. If any of us ran into something that required mathematics beyond our personal skill sets, this fellow was our go-to guy.
He announced to all of us: "I am better than you."
After we recovered from our collective astonishment, one of us asked what he was talking about.
He replied: "How would you get the first derivative of the arctangent function?"
I held up a mathematics textbook.
"I don't need that," he said. "I know it off the top of my head and you don't. That's why I'm better than you."
For the sake of keeping this text fit for family consumption, I won't go any further into the ensuing commentary except to say that it was quite colorful, but wouldn't you know it, I actually found something later on to ask this mathematician about.
I had an eleven pole filter that had been designed into a digital multimeter. I wanted to know if the roots of the eleventh order polynomial of that filter's transfer function could be found; could they be factored out. The answer I got from the mathematician was "no," but he couldn't tell me why that was the case.
In fact, it was the case. The mathematician was right, but I only learned why later from a biographical article in Scientific American about the French mathematician, Évariste Galois (October 25, 1811 – May 31, 1832) who, if I got this right, had sought to find a generalized method of factoring a polynomial of any order and proved that there was no such general method for polynomials of greater than fifth order. Galois' work was the beginning of what is today called "group theory."
Because our resident mathematician was who he was, because of the offensive attitude he displayed, I didn't really believe him. He had lost his credibility with me and as I later saw, with the others too.
There was a life lesson in that.
(John Dunn is an electronics consultant at Ambertec, P.E., P.C. in Merrick, N.Y., a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE) and is a member and former Chairman of The IEEE Consultants Network of Long Island (LICN)).
I had to work with one of those types. Head honcho of SW, nobody could even get an Eprom burned without his approval.
I came up with a good workaround - waited until the officious little twerp went on vacation.
The best way to assure that one gets started on the wrong side of everybody is to announce that "I am better than all of you". That is for certain. The only job that I would not attempt to show somebody like that up on would be the removal of un-exploded ordinance, which is an important job. That would be the task that I would leave to the braggart unchallenged.
Not to go off on the inverse of an arctangent, but the above story reminds me of the joke about the engineer, the physicist, and the mathematician:
A physicist and engineer and a mathematician were sleeping in a hotel room when a fire broke out in one corner of the room. Only the engineer woke up he saw the fire, grabbed a bucket of water and threw it on the fire and the fire went out, then he filled up the bucket again and threw that bucketfull on the ashes as a safety factor, and he went back to sleep. A little later, another fire broke out in a different corner of the room and only the physicist woke up. He went over measured the intensity of the fire, saw what material was burning and went over and carefully measured out exactly 2/3 of a bucket of water and poured it on, putting out the fire perfectly; the physicist went back to sleep. A little later another fire broke out in a different corner of the room. Only the mathematician woke up. He went over looked at the fire, he saw that there was a bucket and he noticed that it had no holes in it; he turned on the faucet and saw that there was water available. He, thus, concluded that there was a solution to the fire problem and he went back to sleep.
Its comforting (but not good) to know that engineers aren't the only ones who can be accused of having the people skills of a porcupine in a crowded swimming pool.
I don't care how good a person is intellectually, if they can't get along with others I'd rather not have them around. A team of good people working together spontaneously is always better than a group of geniuses who don't communicate because they feel others are unworthy of them.
The immediate implication that I take from somebody announcing that they are "better than me" is that they are suffering from a swollen ego. Many career mathematicians are better at math than I am, but few of them are also able to design a circuit and teach a fellow worker how to safely run a lathe, which I can do both. I have worked with some mathematicians who were also good engineers, but a few more who were not. Now, if it had been possible to find one of the eleven roots, then finding the others would have been fairly easy, since, by symmetry, we know that the sum of all n nth roots equals to zero. At least, when expressed as complex numbers, it works that way. No, I am not able to produce a proof for that assertion.
No. The implications of Galois theory (among others) are that roots of polynomials of highest degree 4 have solutions that can be expressed as radicals. Five and above this is impossible (in general*). See for example http://en.wikipedia.org/wiki/Quintic_equation
(*although note the extension to "ultraradicals" by Jerrard and to Hermite's use of Jacobi theta functions and associated elliptic modular functions)
There is also a fascinating old book by Klein, "Lectures on the Icosahedron and the solution of equations of the 5th degree". See also the wiki for icosahedron: http://en.wikipedia.org/wiki/Icosahedron,
from which: "The rotational symmetry group of the regular icosahedron is isomorphic to the alternating group on five letters. This non-abelian simple group is the only non-trivial normal subgroup of the symmetric group on five letters. Since the Galois group of the general quintic equation is isomorphic to the symmetric group on five letters, and this normal subgroup is simple and non-abelian, the general quintic equation does not have a solution in radicals."
So, did you find the poles of the 11th order filter? I thought that's what numerical analysis was far.
Anyway ... most of engineering (or any technical field) comes down to knowing what you don't know, and how quickly you can find it out.
Looking it up in a textbook is not only not cheating, it's expected. Though I suppose these days Google is your friend.