I found an example from electrical engineering of a complex two-dimensional vector space. If you already dislike math, you won't be too happy with these two pages from Circuits, Devices and Systems, © 1976, by Ralph J. Smith, used for a beginning EE class I took in the first summer term of 1977 at the University of Arkansas at Little Rock.
I'm not sure if engineers still prefer to use the letter j instead of the usual letter i in writing a complex number, but they preferred it back in the seventies, so that's what the j is doing here (below), representing the imaginary part of a complex number. Vectors may be written in different equivalent ways, and one of those ways is to use complex numbers, which have a real (x-axis) part and an imaginary (y-axis) part. For example, in the number 8.66 - j5 (see below) the real part is 8.66 and the imaginary part is 5.
What this author says about electrical engineering is also true of quantum mechanics. We have electronic devices (especially iPod, iPhone, iPad, etc) that depend on recently-discovered quantum effects, but in studying how these devices work, and in making the discoveries they are based on, we need abstract ideas. (Very abstract, not to mention formal, ideas! That's what I'm trying to make more sense of in these writings of mine. Quantum theory is too abstract to understand as it stands.)
Yeh, those are my comments in the margin from 36 summers ago concerning TV ownership.
Our vector in this example, below, is called a phasor, a term used in physics as well as engineering, and also in Star Trek, but with a different meaning. Electrical voltages and currents in alternating current (AC) circuits are generally not "in phase" with each other. A phasor has a voltage or current as its "amplitude," and it has a phase angle ("phase" for short). The variation in time of amplitudes and phases of different currents and voltages can be kept track of by the use of phasors.
Yeh, those are my comments in the margin from 36 summers ago concerning TV ownership.
Our vector in this example, below, is called a phasor, a term used in physics as well as engineering, and also in Star Trek, but with a different meaning. Electrical voltages and currents in alternating current (AC) circuits are generally not "in phase" with each other. A phasor has a voltage or current as its "amplitude," and it has a phase angle ("phase" for short). The variation in time of amplitudes and phases of different currents and voltages can be kept track of by the use of phasors.
The "Imaginary" and "Real" axes make this graph (above) a complex two-dimensional vector space. The variation in time of 60 Hz, by the way, is the standard frequency of household AC voltage. Whatever the frequency is, Kenneth, it must be the same for both vectors in order for them to be represented in this simple way.
So, the amplitudes and the phase angles of the two vectors (above) are what are important. And here's the importance of my choosing this example: Amplitude and phase are what are important in quantum mechanics. They take on strange new significance: In quantum measurements, an "amplitude" becomes a "probability amplitude," and phase angles between vectors become those "delicate phase relationships" that are responsible for the weirdness of quantum superposition.
So, the amplitudes and the phase angles of the two vectors (above) are what are important. And here's the importance of my choosing this example: Amplitude and phase are what are important in quantum mechanics. They take on strange new significance: In quantum measurements, an "amplitude" becomes a "probability amplitude," and phase angles between vectors become those "delicate phase relationships" that are responsible for the weirdness of quantum superposition.
But how does this relate to Father's Day of 1977? Well, I was enrolled in the EE class and reading and working problems in this textbook then. Also I'd recently found a job as the 4-pm-to-midnight night watchman at the Old State House in Little Rock. So that's a couple of possible reasons I'm remembering this Father's Day so well.
My family had lunch at The Embers (actually the Plantation Embers), a white-table-cloth restaurant in a better-than-average highway motel called the Pine Bluff Motel. I usually got roast lamb with mint jelly when we ate there, so that's probably what I ate for lunch. And my gift for Dad on that Father's Day was a copy of the New York Times Book Review.
This may be another reason I remember that day better than other Father's Days. I had just discovered that the Book Review by itself was available at Publisher's Bookshop in Little Rock. Back then, it was impossible in Little Rock to get a copy of the New York Times on the day it came out. It took until Tuesday or Wednesday for the Sunday NYT to arrive by truck and be available for sale in LR.
When I handed a copy of the Book Review to Dad at The Embers, I said it was not an expensive gift but also not an easy one to get. He said something like "Is this last week's?" I had the pleasure of saying no, it was the current issue. Whether he knew about the early availability of the Book Review, I don't know. But he sure didn't let on. Dad and I had been to New York together (on our way to Rome) the previous October. Not that he even read the NYT, but he was bookish.
He also got a kick out of the big headline for the leading book review in that June 19, 1977, issue: "Newton, Kant, Ruskin, Virgins All." It was a review of "The Book of Lists". The memorable title of that review could be another reason I recall that day and giving him that as a gift.
P.S. And let's not forget that one scene in A Serious Man takes place at a restaurant called Embers.
P.S. And let's not forget that one scene in A Serious Man takes place at a restaurant called Embers.
