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:

tj wanderlust-words-in-motion

16 minutes ago
In response to "Asleep At the Real"

In antiquity sleep was personified, transcendent, even romantic.

My thoughts? That's easy to say now, but for most of those who lived, even before the industrial revolution, work days were long, food was scarce, and it was hard, strenuous labor. Oh, and forget about weekends! When these people hit that straw pallet I don't know if they fell asleep or just passed out. Just my opinion.

🌖 HuntersMoon

about 9 hours ago
In response to "Asleep At the Real"

Thank you for the rather long discourse on sleep... I'll finish it after my nap.

2 Like

Legerdemain

a day ago
In response to "Let It Lie"

When my oldest began to question if Santa was real, being seven years older than the next child, I asked her what she thought...she said, I don't think he's real. I asked if she was sure. She was. So I told her that Santa is the giving spirit of the season. And now that she was "grown up", she got to do the giving. So she wrapped all her younger brother and sister's presents and kept it secret for years. So, when you stop believing in Santa, you become Santa.

2 Like

Wordsmitty ✍️

a day ago
In response to "Sea Change"

"gaseous Earth geophysicists" - ( snicker, snicker )

A lot of scientists/nerds seem to get that description especially when they get older.

2 Like

🌖 HuntersMoon

a day ago
In response to "Sea Change"

1 Like

tj wanderlust-words-in-motion

a day ago
In response to "Sea Change"

I sea; another solid entry.

2 Like

tj wanderlust-words-in-motion

Jan 26, 2025 at 3:27pm
In response to "Revisited: "Unwritten, Expanded""

I seldom use emojis in texts and spell out words, use correct (I think) punctuation; it is, after all, still writing. But, just recently I was shown a paper written by her fifth-grade daughter for English. The entire story was written in "text-language" complete with emojis (a lot even took the place of words) on a computer and printed out. The child was very proud of the B+ she had gotten on the assignment!

1 Like

Jayne

Jan 26, 2025 at 9:59am
In response to "Revisited: "Unwritten, Expanded""

As someone who does not have, and will not have, TikTok: isn't that vertical by default? Does that skew the overall tolerance for vertical video? (among the masses, not for you 😂)

Stik: I'd agree those get a pass. I've had to do it in an emergency, and you really don't think about turning your phone, especially if you don't shoot a lot of video.

2 Like

Elisa: Snowman Stik

Jan 26, 2025 at 9:39am
In response to "Revisited: "Unwritten, Expanded""

"If I see a vertical video, I ignore it."

Does this apply to videos that call out harmful/discriminatory behavior? I'm willing to give those videos a pass if they're vertical.

2 Like

tj wanderlust-words-in-motion

Jan 25, 2025 at 3:28pm
In response to "Let It Lie"

I cannot say honesty is always the best policy, I write fiction for Pete's sake, or whoever else reads and enjoys it. However, in questions such as, "Does this dress make my butt look big?"

I can honestly answer with, "No." After all, it's not the dress. As for the question, "Is there a Santa Claus?"

"Hell Yeah, I saw him at the mall."

2 Like

January 27, 2025 at 9:52am

#1082856

Sea Change

I'm linking this article from Nautilus, not to instigate any arguments about climate change, but to make a point about science.

Why Our Intuition About Sea-Level Rise Is Wrong
A geologist explains that climate change is not just about a global average sea rise.

The post is quite old by internet standards, having been released about 9 years ago. This is irrelevant to what I have to say about it, but I'm sure the science has advanced somewhat since 2016.

Jerry Mitrovica has been overturning accepted wisdom for decades.

Well, that's one thing science is for. Another thing it's for is to support accepted wisdom, but I don't think that happens as often.

A solid Earth geophysicist at Harvard, he studies the internal structure and processes of the Earth, which has implications for fields from climatology to the timing of human migration and even to the search for life on other planets.

I could look up what "solid Earth geophysicist" is, but it's more fun to joke. Are there also liquid Earth geophysicists and gaseous Earth geophysicists? Does his purview stop at the mantle, where the Earth's guts change from solid to plastic to liquid? If so, what's he doing talking about oceans?

Most of the link is in the form of an interview. The questions part of the interview, I'll put in bold.

What happens when a big glacier like the Greenland Ice Sheet melts?

Three things happen. One is that you’re dumping all of this melt water into the ocean. So the mass of the entire ocean would definitely be going up if ice sheets were melting—as they are today. The second thing that happens is that this gravitational attraction that the ice sheet exerts on the surrounding water diminishes. As a consequence, water migrates away from the ice sheet. The third thing is, as the ice sheet melts, the land underneath the ice sheet pops up; it rebounds.

I think, perhaps, it's only his "second thing" that would seem counterintuitive. I did a link a while back that talked about the Indian Ocean gravitational anomaly, and how the lower gravity there, however tiny, causes sea level to be lower in that region of ocean.

And this is why we have science: because intuition is, very often, wrong, and it can be catastrophically wrong.

Gravity has a very strong effect. So what happens when an ice sheet melts is sea level falls in the vicinity of the melting ice sheet. That is counterintuitive. The question is, how far from the ice sheet do you have to go before the effects of diminished gravity and uplifting crust are small enough that you start to raise sea level? That’s also counterintuitive. It’s 2,000 kilometers away from the ice sheet.

This is another reason why we have science: to quantify these effects.

I know when I was a kid and I learned about tides, I was like "Okay, I get why sea level increases on the side facing the Moon, but what's up with the bulge on the other side?" I didn't fully understand this until I learned about vectors, in college.

Why is the source of the melt important?

If you’re living on the U.S. east coast, or Holland, you don’t need to worry what global average sea-level rise is doing. I was in Holland a few summers ago and was trying to convince the Dutch that if the Greenland ice sheet melts, they have less to worry about than the Antarctic ice sheet melting.

I gotta admit, I struggled with maintaining my acceptance of his credibility as he used Holland for the Netherlands. But I have to remember: he's a geologist, not a geographer. No one can know everything. But I do hope someone clued him in that we don't call the entire country Holland.

Of course, 120,000 years ago, humans weren’t having any impact on climate. That was natural climatic variability.

One could make the argument (as I have in the past) that humans are part of nature, so all the shit we do is also natural. This is technically true, but that doesn't mean it's helpful.

There's a lot more related to sea level change, which is interesting but, again, doesn't really impact my point here. The money quote for me is this one, further down:

So many of your results seem abstract and counterintuitive. Is that a coincidence?

There are so many interesting problems in our science that you can see with your eyes. But your eyes can fool you. Richard Feynman, the great physicist, used to start his physics lectures by showing students their intuition could take them a long way. They could do things just through intuition that would get them roughly the right answer. Then he used to throw some counterintuitive examples at them. Then he said, “This is why you need physics. You need to understand when your intuition might go wrong.” I firmly am a Feynman acolyte. There are some things that you can explain, but as a scientist you’re always going to face things that are counterintuitive.

Most people think of Einstein as the Platonic ideal of genius, but I'm going with Feynman.

Because to see the unexpected is the reward of science.

And, yes, sometimes science itself turns out to be wrong, or incomplete. But it's still better than guesswork or "common sense."

4 Like • Comment • View 3 comments

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