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 13 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 14 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.

May 7, 2020 at 12:36am

#982947

Crossing My Path

I used to believe that we stayed basically the same person all our lives. Then I read something recently, but didn't bother to save a link, about how some scientists did a study on young people, then studied them again when they were older, and what they found that Old You doesn't really have anything in common with Young You. I mean, we've known this from a physical standpoint - cells get replaced and all that - but it kind of calls into question the idea of continuity of consciousness, and identity itself.

PROMPT May 7th

Start your entry today with the words: “I used to believe...”

I did save this link , and it was destined to show up in here at some point. It might as well be now, because I used to have superstitious thoughts, myself, when I was much younger. But I'm a different person now.

The science of superstition – and why people believe in the unbelievable

The number 13, black cats, breaking mirrors, or walking under ladders, may all be things you actively avoid – if you’re anything like the 25% of people in the US who consider themselves superstitious.

Mirrors used to be remarkably expensive, which probably led to social shaming for anyone who broke one, kind of like we shame anyone who flagrantly wastes money. Walking under ladders can be a bad idea for very practical reasons; we tend not to look up, and if there's a ladder, there's a chance someone is on it doing something, and that someone might have slippery fingers, and having something fall on your head is a Bad Idea.

All sorts of explanations have been proposed for the number 13, from the astrological to plain number theory, but none of them are particularly compelling to me. However, there's no goddamn reason to be afraid of black cats. There's one pawing at my face right now. She just wants food. Robin is a good kitty.

Even if you don’t consider yourself a particularly superstitious person, you probably say “bless you” when someone sneezes, just in case the devil should decide to steal their soul – as our ancestors thought possible during a sneeze.

No, I say it because it's de rigeur, a social acknowledgement, like saying "goodbye," even though that literally comes from the phrase "God be with ye." It makes me uncomfortable to do so, though, because I know the origin.

Superstition also explains why many buildings do not have a 13th floor – preferring to label it 14, 14A 12B or M (the 13th letter of the alphabet) on elevator button panels because of concerns about superstitious tenants.

I've stayed in hotels run by and primarily for people of Chinese descent, and those hotels didn't have a fourth floor. As far as a superstition can make any sense at all, that one makes more sense than the "thirteen" thing; from what I understand, the Mandarin word for "four" is very similar to the one for "death." Or something like that, anyway.

Labeling the floor "M" is potentially confusing; that's usually for "Mezzanine," a term I still don't have a firm grip on, but I know that it exists and usually refers to the second floor of the lobby or, in theaters, a middle tier of seating.

On top of this, some airlines such as Air France and Lufthansa, do not have a 13th row. Lufthansa also has no 17th row – because in some countries – such as Italy and Brazil – the typical unlucky number is 17 and not 13.

Prime numbers just don't get enough respect.

Psychologists who have investigated what role superstitions play, have found that they derive from the assumption that a connection exists between co-occurring, non-related events. For instance, the notion that charms promote good luck, or protect you from bad luck.

Also known as "magical thinking."

For many people, engaging with superstitious behaviours provides a sense of control and reduces anxiety – which is why levels of superstition increase at times of stress and angst.

My take? If it makes you feel better, there's nothing inherently wrong with superstition -- provided, of course, no harm is done. The black cat one can directly cause harm, for example, to the cat as well as emotional damage to its human.

But if you insist on wearing the same underpants every time you play the lottery, who cares? Just wash those nasty things occasionally. Unless that's part of your superstition, in which case, get a grip, man. Clearly it isn't working, or you'd be rich by now, and you'd be better off getting a job and wearing clean underwear to it.

Superstitious beliefs have been shown to help promote a positive mental attitude. Although they can lead to irrational decisions, such as trusting in the merits of good luck and destiny rather than sound decision making.

Which is another reason to be wary of superstitions, though it's clear to me that "sound decision making" isn't a widespread human characteristic as it is.

A more traditional illustration is the Curse of the Pharaohs, which is said to be cast upon any person who disturbs the mummy of an Ancient Egyptian person – especially a pharaoh.

While the curse is of course utter nonsense, and we've learned a great deal about ancient humans by studying their dead, those poor bastards were buried or entombed for a reason, which is probably a similar reason to why we bury the dead today. Would you want your bones studied by archaeologists 6,000 years from now? Well, granted, you won't care, because you'll have been dead for 6,000 years, but that just brings up the uncomfortable question of why we do anything if we're just going to be dead in 6,000 years.

Superstitions practices tend to vary across sports, but there are similarities. Within football, gymnastics and athletics, for example, competitors reported praying for success, checking appearance in mirror and dressing well to feel better prepared.

Oh, those are terrible actions, just absolutely terrible. Wait, no they're not. Prayer (or meditation or whatever) can focus the mind, and wanting to look good is pretty damn universal.

So, clearly, superstition isn't all bad. Hell, I consider myself a rationalist, but there are some things I like to do that might be considered superstitious -- because they help me remember my purpose and goals. For example, I have a coin I carry around in my wallet that has sentimental value to me. Every time I open my wallet, I have to be careful not to let it fall out. This reminds me that I try to be careful with money in general, and hopefully keeps me from making stupid spending decisions. Well, most of the time. Well, sometimes. Which is better than none of the time, I suppose. It could easily be mistaken for a "lucky coin," but that's not how I think of it. Usually. Oh, no, now that I've talked about it, the luck will disappear. Dammit!

Still, leave black cats alone. Better yet, adopt one. Other than looking like little house panthers, they're just cats.

Just remember to feed them.

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