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:

Write_Mikey_Write!

about 2 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 4 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 11, 2021 at 12:02am

#1023061

Weekness

I disagree with the headline here. But of course that meant I clicked on it and read the article, so I guess it was effective.

We Live By a Unit of Time That Doesn’t Make Sense
The seven-day week has survived for millennia, despite attempts to make it less chaotic.

Days, months, and years all make sense as units of time—they match up, at least roughly, with the revolutions of Earth, the moon, and the sun.

Days? Sure. Though don't get me started on the difference between a solar day and a sidereal day. (Seriously, don't.)

Years? Absolutely. Though I'd argue that ending/starting the year when we do makes absolutely no sense whatsoever. It's not set to align with anything real and measurable, such as the solstice or perihelion.

Months? No. A lunar month is roughly 29.5 days long. Some calendar months are 31 days (and of course February is blessedly shorter). If months made sense, they'd be 29 or 30 days and they wouldn't fill up a solar year. See Hebrew calendar for an example.

As for the week?

Weeks, however, are much weirder and clunkier. A duration of seven days doesn’t align with any natural cycles or fit cleanly into months or years.

False.

Or, at least, not exactly true. Because, as I noted, a lunar month is 29.5 days, there is roughly one week between major moon phases: new, first quarter (which is actually a half moon, which confused the hell out of Kid Me), full, last quarter. No, it's obviously not exact, but it's way more useful than our completely artificial "months."

Also, 52*7 = 364. Again, not an exact match for a 365-day year, but damn close. Good enough, even. A better match than calendar months to lunar months.

And though the week has been deeply significant to Jews, Christians, and Muslims for centuries, people in many parts of the world happily made do without it, or any other cycles of a similar length, until roughly 150 years ago.

Matter of definition, I think. Again, "people in many parts of the world" still often measured time by lunar phases, which as I noted above, are roughly 1 week long.

For better or worse (I lean toward "worse"), the international standard calendar is the Gregorian, which is essentially Christian (by way of Roman) in origin, so if weeks are important to Christianity (largely, but not entirely, due to the whole Genesis thing), weeks are what the calendar has. But again, Jews use a different calendar, a combination of lunar and solar (which is why Jewish holidays tend to wander compared to our Gregorian system) and from what I've gathered, the Muslim calendar is entirely lunar.

I say the above not to take sides or come down on religion, but to point out that a lot of articles about the calendar, like this one, seem to start from the unstated assumption that the Gregorian calendar is the One True Calendar, which it is not. Being the international standard merely helps us to communicate, especially with global commerce being what it is.

As an analogy, consider the international standard for time: UTC, formerly called Greenwich Mean Time because England happened to be enough of a powerful world force at the time to set the "zero" meridian (Greenwich is basically part of Greater London). But it could be anywhere, and still work. UTC does have one advantage: its inverse, the 180 degree meridian, passes through very few land masses (and yet they still fuck with the International Date Line because humans are weird).

Still, it's completely and entirely arbitrary. Had world history gone differently, the zero meridian could have passed through Paris, Amsterdam, Jerusalem, the Washington Monument, the Jabal al-Nur near Mecca, or the southern tip of Chile. For the purposes of timekeeping, what matters is that there is a standard, not what the standard is.

So it is with the year. But unlike the year, there is no reason to prefer a certain meridian, apart from history and convention. With the year, you could at least start/end it on a solstice, equinox, or apsis.

Now the seven-day week is a global standard—and has come to dominate our sense of where we stand in the flow of time, according to David Henkin, a historian at UC Berkeley. His new book, The Week: A History of the Unnatural Rhythms That Made Us Who We Are, traces the evolution—and analyzes the curious staying power—of what he lovingly refers to as “a recalcitrant calendar unit.”

Sounds like a book I'd actually like to read. And I know I say this every time, but for anyone new here: this is a writing site, so I'm not going to dismiss a source just because it's a thinly-veiled ad for a book.

The week as we know it—a repeating cycle that has seven distinct days and divides work from rest—has been around for about 2,000 years, since ancient Roman times. The Roman week itself blended two separate precedents: One was the Jewish (and later, Christian) Sabbath, which occurred every seven days. The other was a rotation of seven days tracked by timekeepers in the Mediterranean; each day was associated with one of seven celestial bodies (the sun, the moon, and five planets).

I actually have no quibble with this part; it tracks with what I already know. But I will add that "five planets" isn't arbitrary; one can see five planets with the unaided eye: Mercury, Venus, Mars, Jupiter, and Saturn. While the planet just past Saturn, which shall remain nameless here to forestall the inevitable juvenile puns, is sometimes barely visible to people with excellent eyesight, it wasn't recognized as a planet until the age of telescope astronomy.

Also, the discoverer, William Herschel, wanted to name it George.

I'm not kidding.

Incidentally, the element uranium was named after that planet, as it was identified just a few years later. So again, if history had gone slightly differently, we'd all be talking about georgium reactors.

But as usual, I digress. We were talking about the week.

The week has kept its shape since then, but Henkin argues that it has taken on new power in the past 200 years as it has become a tool for coordinating social and commercial plans with ever-widening circles of acquaintances and strangers.

And that's the real kicker here: the week is simply useful. If I know there's a new Star Trek episode every Thursday, I can plan for it. Fortunately, we're no longer tied to broadcast television schedules, so I can watch it whenever I feel like it, but it, like most shows, follows a weekly release schedule.

The rest of the article is a brief interview between the article's writer and the author of the book; it's short and worth reading, but you'll have to go to the site to read it (it might help to open the link in a private window if your browser has that option).

And because it's been a while since I talked about this, here's my second-favorite calendar reform proposal, the Tranquility Calendar

.

My favorite is the one I'm working on that actually returns the year to alignment with solstices and equinoxes, but it has the disadvantage of being damn hard to keep track of, and I still haven't figured out how to handle months. But hey, that's what computers are for, right?

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