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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 24 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

a day 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.

December 16, 2019 at 12:40am

#971532

I'm Not Kidding

As certain as I may seem to be about some things, there's always at least a seed of uncertainty. In some cases, it's more of a sapling, or even a massive tree. So with today's link, I'm eschewing snark and a lot of my usual opinions, and mostly just throwing it out there as food for thought. Yes, I know it's an old article, but the points it brings up are still applicable, given the proliferation of mass shootings in recent times.

http://nautil.us/issue/2/uncertainty/parenthood-the-great-moral-gamble

Parenthood, the Great Moral Gamble

The decision to have a child is more ethically uncertain than you might realize.

So far, I'm in agreement. Parenthood is, in our age, a choice. It's one that I decided on a long time ago: that I did not want to father a child.

I should note, as an aside, that almost every article or think-piece I've ever seen on the choice of parenthood is written from a woman's perspective - this one included. Rarely is a man's opinion on the matter sought or appreciated. I'm aware that there are many men out there who desperately want (or wanted) to become fathers; my own father was one of them (one thing an adopted person can usually say with certainty: I was wanted). I'm also aware that many other men run from the responsibility, and leave the child and its mother in a difficult situation. I don't think much of those men. While I didn't want children, if one fell into my lap (so to speak), I'd accept my share of the responsibility. I don't consider it irresponsible to avoid making that decision in the first place.

Anyway, back to the article. It touches on some philosophical points I encountered recently in another context: that of moral luck, and the dichotomy between intention-based and outcome-based judgements.

I didn’t choose to have a child. Not if “choosing” means something rational—weighing pros and cons, coming to a conclusion. I tried that process but ran away from it because, even though I wanted a child, it seemed to me that creating a whole new person was such an enormity that no one could rationally decide to do such a thing.

English is a funny thing. "I didn't choose to have a child" is, on the surface, semantically equivalent to "I chose not to have a child," but the implications are very different. Later in the article, she clarifies exactly what she meant.

The child might bring happiness to others, or he might ruin people’s lives. It seemed to me that creating life was an act of astonishing hubris because it made me responsible, maybe morally responsible, for huge consequences.

I gotta be honest, here: the moral responsibility issue wasn't a big part of my thought process when I decided I didn't want to be a father. Not that it was something that happened overnight. I didn't wake up one morning and go, "You know what? I don't want to have a kid." It was more of a gradual realization. Lots of considerations went into it, not the least of which was plain laziness; I saw what other men were going through and I didn't want any part of it.

And yet, while it wasn't a big part of my thought process, it was a part of it nonetheless - even though, at the time, I probably couldn't have expressed exactly what I felt about the repercussions. This article does that.

It's a long article and I'm not going to quote much more from it, but it's worth reading, in my opinion (obviously, or I wouldn't have bothered with it).

I'm just going to touch on one of my major bugaboos whenever this sort of discussion comes up:

Cushman speculates that we have a basic instinct to punish bad outcomes arising from a period in our evolutionary history when we were not able to communicate intention reliably, and the best way to encourage pro-social behavior was to punish harm and reward benefit, whether accidental or not.

Can we stop with the fucking evolutionary psychology already? Even if it did matter where our behaviors originated 'way back when, we simply don't have the data to do any more than what the first verb in that sentence says: speculate.

Some years ago, I read a book written (ostensibly; there may have been a ghost) by Jeffrey Dahmer's father. As I'm sure everyone recalls, Dahmer was a notorious serial killer and cannibal who, once caught and incarcerated, died in prison. It's been a long time, so I don't really remember if the senior Dahmer took any responsibility for his son's actions, whether deserved or not.

I think a lot of prospective parents think, if they think about it at all, "maybe my kid will grow up to be the next Einstein." (Or insert someone else glorious in Al's place.) No one I know thinks, "what if the kid grows up to be the next Dahmer?" This is a cognitive bias; one of optimism, perhaps. I'm pretty sure you have about the same number of people with major positive contributions to the world as major negative ones, so the chance is about equal. And there's a far greater chance that the kid would grow up to be just another cog in the machine, probably then begetting more cogs in the machine.

In other words, there's a far greater chance of not winning the lottery than there is of winning it, or of getting hit by lightning.

I chose not to play that lottery.

I don't want to give the impression that I'm judging others' choices. I'm not. But I feel like, in a society as parent-centric as ours is, sometimes I have to justify my decision to others - I've already justified it to myself. Parents, on the other hand, are almost never asked, even implicitly, "what made you decide to have children?" It's just assumed.

That assumption - not an individual's choices - is what I challenge.

3 Like • Comment • View 3 comments

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