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 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.

April 13, 2024 at 2:25am

#1068648

Mawwiage...

Posting early today because it'll be a busy and exhausting day of visiting local wineries with friends.

Can't Buy Me Love: How Romance Wrecked Traditional Marriage

For the sake of context, this 2013 article from Collectors Weekly came out during the time when same-sex marriage was still being debated in the US; it wouldn't be until 2015 that Obergefell v. Hodges settled the matter once and for all. Hey... stop laughing; I said once and for all.

I mention this only because, at the time, "traditional marriage" was a buzzword, and a dog whistle for "white Christian man marries white Christian woman; together they go on to produce 2.1 children, and the man makes all the decisions in the family." And even that is not what was, historically, "traditional marriage."

In reality, it’s the idea of marrying for love that’s untraditional.

This is not even getting into different cultures' traditions.

For most of recorded human history, marriage was an arrangement designed to maximize financial stability.

Much as I'd like to agree with that, it still seems Eurocentric. But, honestly, I don't know enough about other cultures to know all the nuances involved.

By the Middle Ages, gender inequality was not only enshrined in social customs, but also common law. In most European countries, married women were forced to give up control over any personal wealth and property rights to their husbands. Eventually, the system became known as “coverture” (taken from “couverture,” which literally means “coverage” in French), whereby married couples became a single legal entity in which the husband had all power.

One of the more common arguments against same-sex marriage back then was "What's next, you can marry your dog?" To me, that argument told me everything I needed to know about how the person making it would treat women. It's like the idea of "two willing adults wanting to enter into a mutually beneficial agreement" completely escaped them. Once you get to the "two willing adults" hurdle, it's not even a little bit of a stretch to consider that those adults can be any sex and/or gender.

I'd personally be perfectly content to extend that to more than two (for other people, not for me), but that's a fight for another time.

Under such laws, children were generally viewed as assets, in part because they were expected to work for the family business.

Another change: nowadays, they're liabilities. Or, at the very least, it's an emotional bond more than a business arrangement.

Despite the church’s staunch position on monogamy, in the late Middle Ages, a legal marriage was quite easy to obtain. However, as more couples attempted to elope or marry without consent, the old guard upped its game. To combat the spread of “clandestine” marriages, or those unapproved by parents, state officials began wresting the legal process of marriage from the church.

In my view, that sowed the seed that became part of the same-sex marriage debate, at least in the US. Religious people get married twice in the same ceremony: one sealing their bond in the eyes of their religious group, and one making it official to various government agencies, not least of which is the IRS. Much breath was wasted with people talking past each other, not understanding that one person meant religious union, while the other meant civil union.

As this philosophical support for individual choice spread, more young people wanted some say regarding their future spouses. “Demands for consent from the people actually getting married were thought to be quite radical,” says Abbott. Even more radical was the idea that marriage might be entered into for emotional, rather than financial, reasons.

It's also apparently radical that a marriage be considered a partnership between equals.

In fact, for thousands of years, love was mostly seen as a hindrance to marriage, something that would inevitably cause problems. “Most societies have had romantic love, this combination of sexual passion, infatuation, and the romanticization of the partner,” says Coontz. “But very often, those things were seen as inappropriate when attached to marriage. The southern French aristocracy believed that true romantic love was only possible in an adulterous relationship, because marriage was a political, economic, and mercenary event. True love could only exist without it.”

In other words, they knew that love dies, but money is forever.

Anyway, the article goes on for a while, but, as it covers centuries of history, it seems to be a decent summary. It also emphasizes how laws are often slow to catch up to social realities. And yes, there's a nod to the then-current marriage debate.

The laws, if not the attitudes of certain kinds of people, have moved on since then, and we've shifted our focus as a society to trans issues, when we're not contemplating our looming climate apocalypse.

And no, the two have nothing to do with each other. But the shift in attitudes about marriage and the climate problem both seem to have their roots deep in the Industrial Revolution.

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