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 11 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 12 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.

March 31, 2022 at 12:02am

#1029770

The Attempted Resurrection of Words

I've been writing in this blog for a long time, and I usually take steps to avoid duplicating entries -- though I'm willing to bet something has slipped through now and then. Today's article is related to one I did earlier this month (see "Respair!"

), but it's not the same article even if it does have some things in common.

Five life-affirming words we should bring back into use

Unfortunately, none of them have to do with drinking.

Lexicographer and TV personality Susie Dent recently embarked on a curious, self-appointed mission. She is determined to bring the word “respair”, last used around 1525, back into common usage.

This is the point of intersection between this and the previous discussion. Stop trying to make "respair" happen, Susie. It's not going to happen.

There is a direct relationship between a language and the society that uses it. Our needs, beliefs and history are fundamental principles that shape language. Lexicographers have shown that the pandemic has led to an explosion of new words and phrases, including “Blursday” and “covidiot”.

For a long time, I believed, on no real evidence, that some Inuit peoples have dozens of words for "snow." Lots of people believed that. It confirms something in one's own worldview, that language reflects the concerns of the society that uses it.

This turns out to be... kind of not the case. Not really.

The "direct relationship" quoted above sounds to me like some version of the misnamed Sapir-Whorf hypothesis

, which sadly doesn't have anything to do with a certain Klingon serving in Starfleet.

But what do I know? I'm hardly an expert on this sort of thing. I'm no linguist. I'm barely functionally bilingual. Hell, I never even bothered to learn much Klingon.

Here are five terms recorded in the Oxford English Dictionary which are connected, in different ways, with the importance of appreciating and loving oneself, one another and life in general.

If I could be arsed, I'd comb through the OED looking for words related to drinking. My attempt to add my own, danchu, has not yet achieved widespread use (along with several other words I coined). Danchu describes the state of being completely obliterated by drinking, and which later I found has a close approximation in Mandarin that means "fade to black" as in the end of a movie, which was satisfying in My sight.

Adamate: to love very much

This verb is formed on the root of the Latin verb amare, which means “to love”. There is evidence of its use by dramatists in the 17th century.

I mean, we already have "adore" and other words signifying great love. And this one, adamate, is just begging to make puns such as "I see that adamate the apple." Verdict: No.

While it is difficult to establish exactly why “adamate” did not become popular, the more negative associations of the French loan might have played a role.

I can't imagine why. Something like 60% of English words are derived from French, from what I understand, and I'm sure a lot of others have negative associations in French. The English word "excite," for example. In French, the verb exciter means "to arouse," and yes, it means that in the sexual connotation.

Meanwhile, we use "excite" for everything from being pumped to go to a rock concert, down to the smallest shifts in energy levels of an electron.

Autometry: self-measurement, self-estimation

Although still used in mathematics, in connection with measuring the dimensions of something, I am interested here in a single use of “autometry” by the poet Robert Southey. In his 1829 book, Colloquies on the Progress and Prospects of Society, which details imaginary conversations between the author and the social philosopher Thomas More. Southey uses “autometry” to refer to the significance of one’s own judgement: “You judge of others by yourselves,” he writes, “and therefore measure them by an erroneous standard whenever your autometry is false.”

So is it AU-to-met-ry, au-TOM-et-ry, or something else? I don't see the need for this word either, except in some of those writings you see where the author just has to show off his or her enormous vocabulary.

Biophilia: love of life

This word is probably best known as the title of Icelandic singer Björk’s seventh studio album. “Biophilia” and its counterpart “necrophilia” were coined in the 19th century as technical terms in psychology. The popularity of the term “necrophilia” and its increasing association with deviant sexual practices have been boosted by a number of high-profile criminal cases.

Not a Björk fan, and I'd never heard of that album. My thought is people will think it's the antonym of necrophilia, because by default people are always thinking about sex, but the two words have entirely different connotations. But the desire to have sex with the dead is almost universally recognized as deviant, while the desire to have sex with the living is the default.

But no, it just means "love of life." And it uses two well-understood roots: bio- and -philia. So any English (or Greek) speaker encountering the word can instantly guess at that meaning, "necrophilia" notwithstanding. Verdict: whatever.

Collachrymate: to weep together

COVID has of course seen physical proximity severely restricted. In this context, this verb, which represents a physical expression of sympathy, is particularly resonant.

I... I guess I just don't have context for this one. Maybe because of the societal expectations put on men: weeping is something we do alone, and then only when something really bad happens, like your dog dies or your favorite sportsball team loses. I don't think I've ever, in my entire life, even as a child when crying was expected, sought out someone to collachrymate with. Run for comfort to parents, sure, like most kids - but they don't cry with you; they try to make the wailing stop so they can get some damn peace.

While I don't believe in promoting antiquated and useless gender roles, that programming is still with me. Perhaps someday, as a society, we'll overcome it. For those who are inclined to find a weeping partner, I mean, go for it. But will you really want to say, afterwards, "We engaged in collachrymation?"

Verdict: No point.

Mesology: the science of achieving happiness

This noun has been in use in scientific texts since the end of the 19th century. It probably comes from the French word, mésologie, which refers to the study of the relationship between an organism and its environment.

However, we also find the term earlier, around 1830, in the writings of the English philosopher Jeremy Bentham, in what might well be an instance of an ad hoc coinage. He defines “mesology” as the scientific enquiry or branch of logic that deals with the means of attaining happiness.

I've spent some time in here pointing out the futility of having "happiness" as a goal to achieve. The short version is, it's not a goal to strive for, but the result of something else. An emergent property, perhaps.

Therefore, I do not believe that there is a science of achieving happiness. The closest I can think of, for me, is zythology, which is the art and science of beer making. Because beer makes me happy.

Your definition, of course, may vary. And I might also argue that happiness isn't having what you want, but wanting what you have. That's the opposite of setting goals and working toward them.

I find this word distasteful.

In short, how about we find something other than words to give us satisfaction in life?

Unless, of course, you're a linguist, in which case, well, you do you.

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