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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QPdoll

about 7 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 10 hours ago
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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.

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Annette in France

about 21 hours ago
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Bienvenue en France.

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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

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WakeUpAndLive️~Happiness

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

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Elisa: Middle Aged Stik

a day ago
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

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WakeUpAndLive️~Happiness

a day ago
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/

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WakeUpAndLive️~Happiness

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

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

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Aiva Raine

a day ago
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.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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December 21, 2023 at 8:38am

#1061260

Seasoning

Winter solstice today.

Well... it's the winter solstice for the only hemisphere that matters (the northern one); summer in that... lesser hemisphere.

And also well... it's today for me, but tomorrow for, say, Europeans.

This is because, while the solstice occurs at the same moment worldwide, the time depends on what time zone you're in. It's 10:27pm on the 21st, here in Eastern time (WDC time), but 3:27am on the 22nd based on Universal Time (e.g., England).

I've seen some sources claim 3:28am UT / 10:28pm EST. I don't know why there's a one-minute discrepancy there. I'm not sure it matters. Probably some differences in how to round numbers up or down, I'd guess. We're still calculating it with greater accuracy than, say, Stonehenge ever did.

This sort of thing can be confusing, I know. Time zone differences trip me up pretty easily.

Want to be confused more? Great! That's my specialty!

We usually think of the winter solstice as the first day of winter. Okay, that's fair; by the astronomical definition, winter stretches from the winter solstice to the spring equinox (this is true regardless of hemisphere, but for... the unfortunate upside-down people... the winter solstice is in June). But there are other definitions of winter.

Meteorological winter, in contrast, is tied to the Gregorian calendar: December 1 through February 28 (or 29 if it exists) in the northern hemisphere. This roughly centers meteorological winter on what is, on average, usually the coldest period of the year: mid-January. Yes, the average low lags the astronomical season, usually by about a month. Similarly, July here is generally hotter than June.

And then there's solar winter, which is related to but not the same thing as astronomical winter. (I told you this would be confusing.) This definition of winter is largely based on length of daylight, and stretches from around Halloween to Groundhog Day.

Those holidays are not coincidences, either; they derived from much earlier celebrations of the beginning and end of solar winter. This is why you sometimes hear the winter solstice referred to as "midwinter." And the summer solstice has been called "midsummer," as in Shakespeare's play. There are similar days around May 1 and August 1 that mark the solar season transitions or, by the astronomical definition, the middle of each season. (Look. I already said it's confusing. Just roll with it.)

The upshot of all of this is that, while the worst of winter is surely still ahead of us—personally, I dread February the most, but January objectively sucks worse—at least we'll be getting more daylight. The downside of that: more of the accursed daystar every day. Yes, I have a complicated relationship with the sun. I appreciate the warmth, but I do wish it'd stop burning me and glaring at me like I did something wrong.

This, for me, ends the winter holiday season. (As I've noted before, axial tilt is the true reason for the season.) At least until New Year's, which I only acknowledge because it's International Drinking Night.

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