Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 10 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

March 7, 2024 at 9:23am

#1065784

Zoned Out

It's about that time of year for most of us, so here's a timely article from Atlas Obscura (copied from The Conversation) to consider:

Why Daylight Saving Time Messes With Your Brain
To “spring forward” is more damaging to our health than to “fall back,” according to some surprising science.

While health shouldn't be the only measure of a thing's worth, if something's unhealthy, then it at least ought to be fun or provide some other benefit. And I'm not convinced switching clocks around twice a year is of any real benefit.

This bit is US-centric, but other countries use some variant of DST, so it may be relevant.

As people in the U.S. prepare to set their clocks ahead one hour on Sunday, March 10, 2024, I find myself bracing for the annual ritual of media stories about the disruptions to daily routines caused by switching from standard time to daylight saving time.

And now you've added to them. Oops... so have I. Damn.

But the effects go beyond simple inconvenience.

"Inconvenience" is a valid reason to do, or stop doing, something; let's not dismiss it.

Researchers are discovering that “springing ahead” each March is connected with serious negative health effects, including an uptick in heart attacks and teen sleep deprivation.

On the other hand, an unreplicated study or two is not a valid reason to do, or stop doing, something. Mind you, I'm not saying the findings are bullshit, or that I don't like them (I happen to be on the "no time switching" team, personally), just the same sort of thing I've been harping on all week.

I’ve studied the pros and cons of these twice-annual rituals for more than five years as a professor of neurology and pediatrics and the director of Vanderbilt University Medical Center’s sleep division.

Admittedly, those are some impressive credentials.

I do have one quibble, and it's not about neurology, medicine, or somnology:

However, the two time shifts—jolting as they may be—are not equal. Standard time most closely approximates natural light, with the sun directly overhead at or near noon.

This is misleading. Or, well, misleading in most locations. And I'll mansplain why.

First: On a world with 24 equal time zones whose boundaries exactly follow lines of longitude, set up so that the middle longitude of each time zone most closely matches the average zenith of the sun at noon, you still have edge cases: on each edge of the time zone, noon will be off from average solar zenith by 30 minutes. Imagine two people standing on either side of this boundary line of longitude, close enough to touch each other. For one of them, it'll be 11:30 am; for the other, 12:30 pm. And yet both would see the sun at its daily high point in the sky. Well, there would still be a day-to-day wobble, because of the complicated interplay of the Earth's elliptical orbit and its rotation, but hopefully, you get the idea. This offset changes as you cross the time zone going east or west, but only at the middle longitude is it near zero.

Second: For sociopolitical reasons, few areas follow the designated time zones. Some countries (even large ones like China) do away with time zones entirely. This leads to even greater divergence between noon and solar noon at many locations. Hell, some places split the difference (notably areas within Australia and India), and the time there is always something:30 when most of the world sees something:00.

Third: as I noted in "First," solar noon occurs at a different time (based on a standard 24-hour clock) every day, no matter where you're standing. The sun's location in the sky, throughout the year, at any given clock time, moves around not only north to south with the seasons, but also a bit east to west. This results in a very pleasing pattern known as an analemma

(that link, which will take you to Wikipedia, explains the whole thing better than I can in a relatively short blog entry).

I'm focusing on solar noon here because that is traditionally what was meant by "noon:" when a sundial's shadow is shortest during the day.

I recently found an interesting map

that displays the difference between solar time and clock time, by country and time zone. If you go to that link, you'll see a preponderance of red. This does not mean that the country or area in question voted Republican in the last elections; it means that politicians of all colors seem to favor solar time being behind standard time, rather than vice-versa. Note especially Argentina, which should be mostly in the same time zone as the far east of Canada but is not; and China, which, as I said above, only acknowledges one official time.

Other things I found interesting from that map: 1) if you look closely, you can see the 1/2 hour time zone areas; 2) England and France are in different time zones despite being mostly the same longitude (which is terribly on point for both countries); and 3) even London, which through accidents of history defines the basis of all time zones, isn't set for solar noon = clock noon.

Whew. That was a lengthy diversion, especially considering the huge number of people who insist that DST gives them an extra hour of daylight. It does not. It only moves the clock around. Hell, even the end of the article is worded like that.

So I won't quote further from the article; suffice it to say that the author lays out her case along with some fascinating history and statistics. She seems to be on the side of "just adopt standard time year-round," though to that I'd add "bring the time zone boundaries closer to where they ought to be based on UTC."

In closing, then, I'll just add this: it doesn't much matter to me one way or the other, as I'm retired and a night owl. So I don't care on a personal level; I just like things to be logical and consistent, which I know is asking too much of humanity. The most common objection I've heard to year-round DST is: "I don't want to wake up and go to work in the dark." That argument is a condemnation of capitalism and the Protestant work ethic, not a reason to keep switching clocks around.

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