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:

Wordsmitty ✍️

about 11 hours ago
In response to "Penultimatum"

It's been an enlightening run. We wish you well in your next adventure.

It will be interesting to see if after the next entry that WDC actually does stop you from adding another.

We're all waiting on pens and pencils for the unveiling of the new blog and to discover what it will be called.

1 Like

BIG BAD WOLF is Merry

about 14 hours ago
In response to "Penultimatum"

I'm reminded of this scene from The Princess Bride.

(Vizzini has just cut the rope The Dread Pirate Roberts is climbing up)
Vizzini: HE DIDN'T FALL? INCONCEIVABLE.
Inigo Montoya: You keep using that word. I do not think it means what you think it means.

2 Like

Soldier_Mike

a day ago
In response to "Revisited: "In Spite Of""

Tangentially related to 6. Adidas and Puma: During my first tour in Germany, I was assigned to the Army agency that formerly occupied Herzo Base, an airfield probably located within medium mortar range from the sport shoe factories.

As far as German revenge is concerned, I would venture a guess that it's planned down to the smallest detail and executed with appropriate gusto.

tj wanderlust-words-in-motion

a day ago
In response to "Automatic for the People"

Earlier today I did a bit of shopping. I found a couple of items I had not planned on purchasing on sale so I grabbed them. As the cashier rang them up, I noticed they did not register the sale price, the regular price, but an even higher price. I questioned it and was told this, "The sale ended yesterday, but we didn't get the sign changed. Today the price increased because our cost has now increased." Yes, how did you guess, one of the items was eggs, now over a dollar each!

1 Like

StaiNed

a day ago
In response to "Automatic for the People"

I work in a grocery store here in Canada, and agree with a lot.

1 Like

Soldier_Mike

Mar 8, 2025 at 11:59pm
In response to "Story Time"

Causation vs correlation. Reminds me of the phrases we've all seen here on site with respect to comma usage (e.g. Come down and eat, children! vs Come down and eat children!).

1 Like

tj wanderlust-words-in-motion

Mar 8, 2025 at 12:34pm
In response to "Just Teaming"

What about "teamMe"!

When I hear "team player" I also hear the boss expecting everyone to give 110%, then going back to the office instead of pitching it.

2 Like

🌖 HuntersMoon

Mar 8, 2025 at 9:32am
In response to "Just Teaming"

4 Like

tj wanderlust-words-in-motion

Mar 7, 2025 at 4:11pm
In response to "One Fine Thing"

Soldier_Mike

recently I asked on of my grandchildren if she new what an octothorpe was (she uses hashtag) and after a minute of deep thought replied, "A mythical creature."

1 Like

Soldier_Mike

Mar 7, 2025 at 3:17pm
In response to "One Fine Thing"

Regarding your Exclaimer!, I never heard the term octothorpe before; it was, as you point out, either pound or number (sign) in my experience. As an aside, it's only visually similar to, not identical to and interchangeable with, the musical sharp sign. The octothorpe's left-to-right lines are horizontal, while the similar lines on the sharp sign are slightly angled from lower left to upper right (# vs ♯).

2 Like

March 11, 2025 at 9:45am

#1085201

Penultimatum

Interesting word, 'penultimate.' As the Latinized synonym for "next-to-last" or "last but one," it is, like most Latinizations, considered more erudite, perspicacious, or even loquacious than the vulgar Anglo-Saxonisms that mean the same thing.

But it's also sometimes misunderstood, which means it either needs to be explained, put in proper context, or avoided entirely (the latter of which will do nothing to mitigate its misunderstoodiness). I don't say this to rag on anyone or try to make them feel stupid, not like when I rail against misuse of apostrophes. But I've seen it used as if the writer thought it meant something like "beyond ultimate," which is just nonsense, like giving 110%; or, alternatively, in the same spirit as 'inflammable,' that it's just a wordier synonym, like 'utilize' for 'use.'

That's not the odd thing, though. As I've noted in the past, we all start out ignorant and have to learn things, such as words and what they mean. That's okay, as long as one doesn't cling to the mistake out of pride or stubbornness. No, what's odd is that, unlike other words, the wrong usage has not (yet) taken off and become standard.

'Decimate,' for instance, is meant to describe the loss of 1/10th (or 10%, same thing) of something. It comes from Latin, again, naturally, and shares a root with 'decimal' (hence the 1/10). Supposedly, a legion that needed to be punished would be purposely decimated in order to keep the remaining 90% on their toes, much like layoffs today are meant to strike fear into the hearts of uppity workers, lest they labor under the misunderstanding that they have any power whatsoever.

But so many people have misused the word that it's now a synonym for the similar-sounding 'devastate,' and it drops the 'decimal' connection completely, losing its reference to any particular number or fraction. Any group or area that suffers significant losses, be it a tenth or half or whatever, is said to be decimated.

I said it was wrong, up there, but it's not, really; it's just an example of how language changes. Look up 'decimate' in a dictionary and you might find reference to its old meaning, but it's clear that the English word has fully embraced its new, less numerically precise, definition.

Not that we should look to dictionaries to resolve arguments. As I've noted repeatedly, they're descriptive, not prescriptive; that is, they describe how language is widely used, rather than prescribe how language should be used. If enough of us pedants rose up in protest and demanded that 'decimate' be used in the Latin sense, they'd have to add that definition back to the dictionaries.

Point being, I don't know what threshold the wrong use of a word or phrase has to reach before becoming accepted to all but the most dedicated language police. But it seems 'penultimate' hasn't reached that threshold, and by using it to describe the second-to-last entry in this blog (that is, this one), I hope I'm doing my part to keep the definition clean.

Tomorrow, then, I say goodbye.

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