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:

Turkey DrumStik

about 5 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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').

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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

December 5, 2018 at 12:45am

#946869

It's Alive!

I am fortunate in that, most days, I can set my own sleep schedule. Most people can't, due to work, kids or other obligations.

Previous entries have gone into more detail about this, but left to my own devices as I usually am, I'm basically biphasic, sleeping twice in a 24 hour period.

I've long believed, without any real evidence to support this, that everyone has their own sleep preferences, and that to disturb these sleep cycles leads to more stress. Well, now I might have some evidence.

https://getpocket.com/explore/item/if-you-re-just-not-a-morning-person-science-s...

I've seen some of these assertions before - that humans have a greater-than-24-hour natural cycle, for instance. I recall one study that left people in a windowless, indoor environment without clocks for an extended period, and let them sleep and wake on their own. I think the study concluded the natural cycle is about 24.5 hours. Which is weird, because a) the earth's rotation is gradually slowing down due to tidal friction from the moon, so if anything we should have inherited a less-than-24-hour cycle from our distant ancestors, and b) Mars' day is approximately 24.5 hours, and we're probably not Martians. So I don't know what's up with that; as far as I know, no one does.

What's not controversial is that there are individual differences from the "standard" sleep cycle of 11pm-7am or thereabouts. I was on such a schedule for most of my working life, and it messed with me. The times when I could sleep past 7 were rare, but I always felt that those were the occasions when I got my best sleep. Further, trying to fall asleep around 11pm was a chore for me.

I still have questions, though, such as: what happens when someone with a nonstandard monophasic cycle switches time zones? Say, moving to another longitude. We know that jet lag is a thing, but usually it's overcome in a few days of adjusting one's internal clock to local time. But can this be used to adjust, say, someone who sleeps from 2-10 in one time zone to adapt to an 11-7 schedule in another, without too much stress?

Seasonal variations in daylight hours almost certainly play a role in this as well. Right now where I am, it gets dark around 5; in the summer, during DST, darkness might not occur until 9pm, with sunrise coming earlier (I don't know offhand when sunrise happens, for the obvious reason that I'm asleep at the time). And we know that many people, myself included, "enjoy" seasonal mood swings related to the amount of light we get from the accursed daystar.

But probably the worst thing about being a night owl is suffering the scorn of morning people. I used to vacation in the Outer Banks of North Carolina every year, with a bunch of friends, and I was usually the last one to go to sleep and the last to awaken (I don't believe in keeping strict time schedules on vacation). Inevitably, I'd get up around the crack of noon, and everyone else would be all sarcastic with their "Morning, sunshine" and "It's alive!" It got on my nerves, and then my friends wondered why I'm antisocial.

Now, I could probably fix some of that by kicking the caffeine habit, but I don't really want to. In fact, I tried it a few times when I was working, but all it did was make me grumpier and give me a headache, so these trials never lasted very long. Perhaps now, when I can pretty much set my own schedule, I could try tapering off on the wakey-wakey juice to see what my unmedicated cycle might be like. But honestly, for the same reason that I can, I don't feel a pressing need to do so. I have far more urgent habits to revise, such as eating better and getting more exercise. Fortunately, there's a 24-hour gym nearby. Unfortunately, I can't be arsed to go to it at any time. Too many other things to do.

So I'll just urge everyone here to recognize that sleeping late is in no way a moral failing; it's natural for many people. Leave us alone and can the snark. We'll get up early if there's something to do; otherwise, let us sleep and don't give us shit about it.

4 Like • Comment • View 5 comments

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