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:

tj-Merry Mischief Maker

about 8 hours ago
In response to "Extro Special"

I'll also use the spelling extrovert because if I use extravert my spell checker flags it. In college psychology our professor termed his own words, innervert and outervert because he thought they fit better. I, however, thought they sounded like definitions of belly buttons.

1 Like

Kåre เลียม Enga

about 14 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

Dec 26, 2024 at 4:20pm
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

December 28, 2024 at 8:53am

#1081623

Extro Special

SciAm takes on a spelling challenge. Or, it did, nine years ago. Well, actually, it was an opinion piece even then.

The Difference between ExtrAversion and ExtrOversion
What's the correct spelling: ExtrAversion or ExtrOversion?

Let's find out what they say, then, if we can crawl out of our introvert holes long enough to give them a look.

Jung may be rolling in his grave.

Lots of people are. I propose wrapping them in copper wire, installing some magnets, and turning them into power generators.

Folklore has it that when Carl Jung was once asked which was the correct spelling—ExtrAvert or ExtrOvert—Jung's secretary wrote back something like, "Dr. Jung says it's ExtrAverted, because ExtrOverted is just bad latin."

That's rich, claiming folkore in a story about Jung.

The thing about Latin is, as a dead language, all of its rules are set in stone. When people actually spoke it, though, it was widespread enough that it changed over time, and we, at some point, decided that the Latin in administrative use around Julius Caesar's time (if I recall all this correctly) was the Latin, and the usages and spellings were calcified. In reality, people went on speaking and writing it, and it eventually morphed into Italian, French, Spanish, etc.

If it helps, the French translation of the adjective extraverted is extraverti(e).

But. This is English. A very widespread living language, subject to change and regional variations. What's the correct way to spell humour? Gray? Tire? Kerb? Once something gets set loose in the public, at some point, it stops being a mistake and starts being a variant. Yes, sometimes I rail against those variants, but I have to remind myself that I'm witnessing a linguistic shift as it occurs.

It's always a mistake to use it's as a possessive pronoun, though.

One of the first times Carl Jung introduced the term is in 1917, in his book "Die Psychologie der Unbewussten Prozesse", he spelled it "ExtrAvert". Exhibit A (ha ha):

You'll have to go to the link to see the example, because it's a graphic, but please note that it's in German. English is about as much German as it is French, and neither language controls English spelling.

So why do so many people spell it ExtrOversion today?

At this point in my first reading of the article, I took a wild guess: to conform better with the spelling of its antonym, introvert.

The article then dates the English "o" spelling to one Phyllis Blanchard in 1918, which, as you might note, is but one year after Jung's book above. It also specifies American English, which, as we all know from the above examples I provided, need not conform to other Anglophone countries' spelling.

Not only did she change the spelling of the word, but she also changed the definition!

Definitions, too, change over time and culture. You know what the French translation of the English verb "to request" is? "Demander." This is, of course, cognate with another word in English that has a much stronger connotation than "request."

What I think probably happened is that she was translating Jung and used the "extro" form to imitate the "intro" form for symmetry.

That is, as the author admits, a guess. But it was my guess, too.

We now know that there are five fundamental dimensions of personality (extraversion, neuroticism, conscientiousness, agreeableness, and intellect/imagination), each one on a continuum.

Yeah, as with any other psychological "knowledge," this, too, is subject to alteration over time. But usually, it changes in a somewhat more technical manner than spelling or everyday word usage.

Under this framework, extraversion is defined as being outgoing, sociable, expressive, and assertive. Introversion is defined as the opposite of extraversion (reserved, quiet).

Anyone who's met me can tell you I'm not very reserved or quiet, and yet I don't identify as an extrovert.

Why does this matter? Trust me, I'm not usually this pedantic.

But I am. Being an introvert and all.

Maybe a solution is to just have both spellings in existence, but define the terms differently.

Oh, gods, no, no, no, please don't. I went down a rabbit hole recently on the difference between kluge and kludge (in the process discovering that neither one is actually of Yiddish origin, much to my disappointment), and, well, let's just say I left the rabbit hole even more confused than when I fell into it.

How about instead we bury ExtrOversion once and for all, and all embrace the same spelling to honor Jung.

The author, in passages I didn't quote, appears to be a massive fan of Jung. I can kind of understand this (he was better than Freud, at the very least), but, again, he wrote in German. Which is a fine language, but, as I said, has no direct connection to English (which I've come to understand as a mature creole of earlier forms of French and German).

Fortunately, the author ends with a sentiment I can certainly support:

I do believe it's helpful for scientists to listen to the experiences of individuals, but I also think it could be helpful for individuals to listen to the latest science.

Science, however, does not and should not dictate word spellings. Hell, they can't even dictate word usage; "theory," for example, means something completely different to a scientist than it does to an ordinary person. And yet, I'm going to continue to spell it "extrovert," because it's totally acceptable in English, if not Latin (or German). If you're more familiar than I am with the word in other Anglophone countries, or even in other languages, feel free to chime in.

3 Like • Comment • View 2 comments

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