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:

Write_Mikey_Write!

a day 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

a day 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.

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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