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:

Writer_Mike

about 13 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

a day ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

a day ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

Sep 21, 2024 at 7:34am
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:46am
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:16am
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

January 11, 2023 at 1:47am

#1042949

Revisited: "Eruv L'olam"

I am fatigued from my journey, though at least my good fortune held out. The trip home was entirely uneventful apart from, again, some minor delays for construction. There's a reason why I use the phrase "as sure as construction on the Capital Beltway" to describe a thing that is absolutely certain to happen.

I even found street parking in Manhattan, which has never happened before. More, the spot was right across the street from my cousin's apartment building. I triple-checked; it was a legal parking space, though only until 9:30 am yesterday. Fortunately, I'd planned on leaving before then, anyway. I'm still waiting for the universe to throw me a curve ball to compensate. Watch this space; I'm sure it'll happen soon.

So it'll be tomorrow before things get back to what passes for normal here in the blog. Since the trip was so ordinary, I had little to write about today, so I decided to use the Wayback Machine, and behold, the entry it landed on was "Eruv L'olam"

from early August, 2020.

That month was an off-month for 30DBC, with the theme of travel, which, going in, I thought would be a good fit for me. Couldn't have been more wrong, as it started off in Antarctica. I can only reiterate what I said about it at the time: Nope.

So instead of participating in that, I did my usual article-linking. In yet another example of cosmic serendipity, the article (which still exists

as of today) concerned... Manhattan.

More specifically, it concerns the Jewish population of Manhattan, which, if memory serves, is higher than the Jewish population of Jerusalem.

Now, a bit of a personal aside: I make no secret of my heritage, as anyone who reads here regularly knows. But to me, that's all it is, a heritage. My life is almost entirely secular. I'm aware of Jewish holidays, but I don't observe any of them. Well, I have been known to make latkes on Hanukkah, but as I've said before, everyone's Jewish for latkes.

What makes a person Jewish can be complicated. Essentially, by tradition, a person is Jewish if their mother is Jewish. This matrilineal descent is rare in the West. I think other groups practice it, but there's a strong focus on paternal inheritance. This is most obvious when you consider how many WASP men have Jr or III after their name, signifying a continuation of one's father's line.

Reality is messy, however, and I think it was the absurdist playwright Bertolt Brecht (not a Jew, but fled Nazi Germany anyway) who pointed out that a father can never be completely sure that a child is his, while the mother usually is (I say "usually" because there have been recorded instances of inadvertent baby-switching at hospitals, including at the one I use, which implies that there were probably unrecorded instances as well).

This is not meant as a slight against women in general, incidentally. Lots of people raise children that aren't biologically theirs. My parents were two of them. Which leads me to the Talmudic question of whether I'm "really" Jewish or not.

The matrilineal descent thing doesn't help me there. My mother was Jewish, but she wasn't my biological mother. I don't know, and don't care to know, anything about my birth mother, but as she lived in the Midwest, I find it highly unlikely that she was Jewish. But I was adopted shortly after birth, and raised Jewish in a Jewish household and even went through a Bar Mitzvah. On the other hand, my religious life ended many years ago, so it's more like a fourth-generation American tracing their ancestry to, say, Scotland, calling themselves Scottish.

I even asked a rabbi about this conundrum once. He thought about it, shrugged, and said "I don't know."

I'm sure other rabbis would have different opinions. Being able to say "I don't know" is, however, the beginning of, if not the very soul of, wisdom, so I will always accept that answer. It's very scientific.

Just to be clear, while I still find the subject of that 2020 blog entry interesting (and still believe that my solution to it would change the world, or at least the tiny percentage of it which is Jewish), it's mostly an academic thing to me, like learning about quantum physics: it makes no difference whatsoever to my life. Even when I was a kid, we weren't the kind of Jews who were careful about doing / not doing certain things on Shabbat. Hell, we lived out in the country; the only way to get to Shabbat services was to drive, which is a big no-no for the Orthodox. Which is why so many Orthodox Jews in the US live in cities like New York, and preferably within walking distance of a shul.

Me, I only care about living within walking distance of a bar. Which I do. Several, in fact. Not as close as if I were living in New York, but at least I can afford to live where I do. Manhattan is bloody expensive.

If that sounds irreligious to you, well, I am irreligious. But even if I weren't, keep in mind that the Jews have a holiday where the whole purpose is to get so completely danchu that you can't tell the good guys from the bad guys. Even if that's a bit of an exaggeration, I wasn't raised with the whole Puritan anti-drinking ethos. I may take it a bit too far sometimes, but consider that a compensation for living in a country that's still reeling from the aftershocks of Prohibition, 100 years after.

As for that referenced blog entry, I even found a way to relate it to the Antarctica theme of the 30DBC prompt. Damn, I'm good.

Oh, and one more thing: I never did explain that entry's title. "Eruv" is explained in the text and in the linked article. The rest of the story is that "L'olam" is the transliteration of a Hebrew word meaning "of the world." Or, in modern usage, it can also mean "of the universe."

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