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 5 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 7 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').

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 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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