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 21 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 23 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').

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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

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

January 20, 2020 at 12:53am

#973758

Don't Fear The Reaper

PROMPT January 20th

Today is a national holiday is the US: Dr Martin Luther King Jr Day. Dr King believed that fear was the ultimate cause of hatred, prejudice, and violence. Do you agree or disagree with his assessment? Give examples.

Lots of people certainly think so. The "-phobia" suffix literally means "fear," but it's been hijacked into service to mean "hatred" as well: homophobia, Islamophobia, transphobia. I don't think that people who harbor a distaste of different lifestyles or religions necessarily fear them, so I object on general principles to conflating hate and fear.

Once, many years ago, when I was more social, I attended a meditation class where the leader would give a little sermon kind of thing before the meditation part, kind of like church I guess. Most of them slid right out of my head, but one homily that I remember to this day: "There are only two emotions: love and fear. All other emotions boil down to one of those."

I found out later that this was stolen from a book called A Course In Miracles, which from what I understand (never read it) is one of those new-agey self-help type books. But I'm not one to accept or reject an idea just because of its source; as they say, even though in our digital age this cliché is becoming increasingly irrelevant, "even a stopped clock is right twice a day."

So I gave it some thought. Emotions don't lend themselves well to a scientific approach, I think, but eventually I came to the conclusion that the love/fear axis is certainly one way to categorize them, if we can categorize them at all as if they all belonged to either the plant kingdom or the animal kingdom.

Typing this now, I vaguely recall having this discussion in a prior blog entry, but I can't be arsed to go look it up. In any case, it's been a while, so maybe I have new perspectives on the idea, a different approach to its narrative. Perhaps if you go digging you'll find it. There will probably be contradictions between this entry and that one. It happens. Memory isn't like a videotape; it's more like a book that you never stop editing and sometimes can't remember what chapter you're working on or even who the main characters are.

But I digress. Love and fear. Under "love" in this system, I suppose, fall all of those emotions that we consider positive or desirable: joy, satisfaction, affection, whatever. I'm no expert on emotions or naming them. Conversely, then, "fear" would encompass all of those negative, dark-side emotions such as anger, jealousy, and, yes... hatred. I'm sure there are emotions that don't fit neatly into one category or the other, though; "amusement," for one, seems positive and yet it often has a tragic source; much of humor is based on some misfortune or embarrassment or even injury or death. Probably only tangential to the discussion at hand, though, so I'll table that one.

But in trying to reconcile this world-view with my own (which, to be honest, isn't very emotion-centric at all), it occurred to me that, if you're going to categorize emotions this way, you can go one step further: fear is, after all, a real or perceived threat to someone or something that you love (possibly including yourself), and so the only emotion, the mother and father and ultimate source of them all, is love.

People are welcome to push back on this idea. After all, what I'm saying is this: the "ultimate cause of hatred, prejudice and violence" is love. Should I not be called out on that? Should I not be kicked back into my dark, forsaken hermit cave for daring to tar the sacred name of Love? Maybe you feel a bit of, dare I say it, anger that I could make such a blasphemous assertion? If so - and it's okay if you do - perhaps it's because you love your world-view and your own particular thoughts and feelings about emotions, and what I'm saying here might be something of a threat?

If, on the other hand, what I'm saying doesn't threaten your world-view, you might not feel any emotion about it at all. Maybe you're just rolling your eyes and shaking your head like a teenager embarrassed by her parents. But hey - there's love in that scenario, too, isn't there? You can't be embarrassed by the antics of pure, indifferent strangers, after all.

Still, how can love be the source of hate? That seems contradictory, because those are often considered opposite emotions (though as I imply in the previous paragraph, I believe the true opposite of love is indifference). Well, let's take the kind of hate MLKJr was talking about: racism. I'm probably going to tread on sacred ground here, too, but bear with me, if you would; I really am leading to something.

My country's Original Sin was racism: first directed at the indigenous population, and then at the African slaves brought over against their will, and, at various times, against the Irish, the Italians, Asians, Muslims and of course always the Jews. While nominally founded on the idea of equality for all, we've always fallen short in practice. The particular brand of racism of white-on-black metastasized into hatred, the ironically-named Civil War, institutionalized segregation, and lynchings and the like - all bearing the same resemblance to "love" as a pile of shit does to a cute puppy.

Nevertheless, just as it's possible, in the extreme case, for one to love only oneself (narcissism), it's also possible to love only one's own family, not giving half a damn about others. A step up from that, I suppose, would be to love one's own clan or tribe to the exclusion of all others, or even preferring one's own perceived race to that of others (I say "perceived" because "race" is ill-defined). This kind of exclusionary love is still love - but it contains within it the darkness of hatred and, yes, fear.

Fear, because anything different, however you may define "different," is a threat - real or perceived - to the things that you love. We fear death, for instance, because we love life.

I sense that I've gone on too long, now, and I won't flog the point any further. I will say that, once I came to this realization - as I come to most of my epiphanies, whilst driving alone at night - I changed my own relationship to love and fear. I can't say that I'm fearless, of course; that would be idiotic and a lie, but I can say it doesn't have the same grip on me that it used to.

I will note, in closing, that you might have heard about a rally that's going to take place in my state capital, once the capital of the Confederacy, today. The purpose of the rally is ostensibly gun rights, an issue that, in the US, is charged with racism (if you don't believe me, look at the now-defunct NRA Channel, and count the number of people of color featured in their programs - or if you can't bring yourself to look into the wayback machine, I'll provide a spoiler: zero, or close enough to it as to be statistically identical). Now, I don't live in that city, though mine has not been without its international news coverage recently; "Charlottesville" has become a household name for all the wrong reasons, as all of the bad actors here were imported from other places, other states even.

It's no accident, I think, that they chose MLKJr day for their protest. Now, this is the US, and most particularly Virginia, so I'll defend their right to protest and be heard, regardless of whether I agree with them or not, but a lot of people are expecting this to turn ugly.

What probably didn't make national news, though, is that last week, in that very same city, Virginia became the key 38th state to ratify the Equal Rights Amendment. This probably has no more than symbolic value, as that particular Constitutional amendment was proposed back in the early 70s and some legal experts assert that it has expired. But symbols have power, regardless, and whatever happens today, I wanted to contrast the negative with the positive.

Fear is real. Hatred, prejudice, and violence are all part of the human condition. But so is love, acceptance, and the desire to become something better.

And cowbell. Lots and lots of cowbell.

Love of two is one
Here but now they're gone
Came the last night of sadness
And it was clear she couldn't go on
Then the door was open and the wind appeared
The candles blew then disappeared
The curtains flew then he appeared, saying don't be afraid

Come on baby, and she had no fear
And she ran to him, then they started to fly
They looked backward and said goodby, she had become like they are
She had taken his hand, she had become like they are
Come on baby, don't fear the reaper

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