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:

Annette in France

about 9 hours ago
In response to "It's Hot in The South, Once Again!"

The quote is too small and low contrast to be read.

Elycia Lee ☮

about 19 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

a day ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

a day ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

Sep 25, 2024 at 7:53pm
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

Sep 25, 2024 at 3:20pm
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

Sep 25, 2024 at 2:49pm
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

August 13, 2019 at 12:26am

#964132

Write Angles

https://www.inc.com/christina-desmarais/43-embarrassing-phrases-even-smart-peopl...

English doesn't make a lot of sense. I'd like to be able to say that this is by design, but no one "designed" English. It's what you get when a bunch of other languages get into a bar brawl with no clear winner, while drunk.

The upside of this is that people who can make any sense at all out of it get the privilege of feeling superior.

When someone uses grammar incorrectly do you make an assumption about his or her intelligence or education?

Yep.

For all the people who insist that "spelling doesn't matter" and "you're being pedantic if you care about grammar," I make an effort. I'm not always successful - few are - but at least I try.

I'm supposed to be the lazy one, here.

On the other paw, spelling and grammar change organically over time, and some changes have good reasons behind them. Anyone who screams at the highway department for signs like "No Thru Street" has Issues. Everyone knows what is meant, and you've only got a short time to look at that sign; it just makes sense to shorten the word from "through," which could be confusing as it looks like "trough," "tough," "thorough," "though," etc. - all of which are also pronounced differently, just for funsies.

In short, it's okay to make changes I'm okay with, and it's not okay to make changes that piss me off. I never said I had to be consistent; hell, English isn't.

Anyway, the article.

Like it or not, words, spelling, and punctuation are powerful and can leave a lasting impression on others. But even the most educated people often unknowingly make common writing and speaking flubs. Check out this long list of ubiquitous grammar mistakes.

I'm not going to copy most of them here; I'm just going to note the ones I take issue with.

4. "I" as the last word in a sentence.

Yeah... that's too simplistic. I can think of examples where a sentence should end in the first person singular pronoun. "There but for the grace of God go I" is a pretty famous one. The real trick is to figure out whether the pronoun is a subject (I) or object (me); the former is rarely placed as the last word, but it does happen.

5. "Me" as the first word in a sentence.

Okay, sure, this is technically wrong from a purist point of view, but I grew up in the South, and that's just how we talk.

24. Hot water heater

If anything, it's a cold water heater. Just use "water heater."

Except for the edge case when you have a new water heater and it fills up with cool water, the water in the tank is already hot when the gas, or heating element, kicks in and makes it a bit hotter. Okay, maybe also if you live in a shitty apartment and you use up all the hot water in a five-minute shower. Point is, I wouldn't ding anyone for saying "hot water heater," even though "water heater" is more appropriate.

28. Subject and pronoun disagreement.

This one is subject to debate...

Yeah, it really is. Despite numerous attempts, no one has been able to come up with a genderless singular pronoun that's widely accepted. (Don't get me started on the idiotic "xir" bullshit. Nothing natural in English starts with x. Before you say "xylophone" or "xenophobia," those are from Greek roots.) Until then, if the antecedent is of indeterminate or nonbinary gender, by all means, use "they/their" unless they (see what I did there) have made their (again) preferred pronouns plain.

34. Throws of passion

Just know that a throe is a sharp attack of emotion. So, to be in the "throes of passion" is to be violently consumed by something.

I'm highlighting this one because of the mental image I got of "throws of passion."

Again, it's not that I never make mistakes. I just think that, as writers, we owe it to ourselves and our readers to do the best we can, and keep learning this weird-ass language.

2 Like • Comment

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