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!

about 14 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 16 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.

April 30, 2020 at 12:03am

#982361

Regrets

Old article is old, but what the hell; I'm doing "30-Day Blogging Challenge ON HIATUS"

[13+] again starting tomorrow and this will probably be the last of these for a month. Unless I foul up 30DBC, which is always a possibility.

https://www.mentalfloss.com/article/67288/10-inventors-who-came-regret-their-cre...

10 Inventors Who Came to Regret Their Creations
Just because someone's invented something, it doesn't mean that they're happy with the end result.

Unlike Cracked, this site apparently counts up instead of down. This takes some getting used to.

1. J. Robert Oppenheimer/ Albert Einstein — The atomic bomb.

Despite past associations with left wing organizations, Oppenheimer welcomed the opportunity to play a part in the war effort. Later, however, he had mixed feelings about the bomb.

But hey, he was also responsible (along with Civ IV) for introducing the West to the Bhagavad Gita

, so there's that.

Einstein was less equivocal.

In retrospect, on balance, I think Einstein probably did more good than harm.

2. Mikhail Kalashnikov — AK-47.

"I keep coming back to the same questions. If my rifle claimed people’s lives, can it be that I…, an Orthodox believer, am to blame for their deaths, even if they are my enemies?" he wrote in a letter to the head of the Russian Orthodox church in 2010.

I skipped quoting a sentence of dubious grammatical provenance there. Pedantry aside, though, between this and the first entry, we should have learned by now that no weapon will bring peace in the long term. Or, well... they all bring peace, depending upon your point of view.

3. Tim Berners Lee — the double slash.

"Really, if you think about it, it doesn't need the //. I could have designed it not to have the //," he said, according to Business Insider.

Yes, the internet is a good thing in general. No, the double slash found in web addresses is far from its worst feature, since these days, no one ever has to actually type it.

4. Ethan Zuckerman — the pop-up advert.

If you've ever found yourself yelling at your computer screen in frustration as yet another pop-up ad leaps into view, obscuring the content behind it, Zuckerman is the person to blame.

WAY worse than the double slash. I wish there were a Hell so that this guy could burn in it for all eternity. On the other hand, if he hadn't invented it, someone else would have.

5. Dong Nguyen — Flappy Bird.

Flappy Bird was a sensation a year ago.

Again, old article is old. I vaguely remember Flappy Bird being a brief flare in the cosmos of the internet. Meh.

6. Bob Propst — the office cubicle.

While working as a consultant for Herman Miller in the 1960s, Bob Propst introduced America to the open plan office and with it, the office cubicle.

This guy should share cubicle space in Hell with Zuckerman.

7. Vincent Connare — Comic Sans.

"If you love it, you don't know much about typography." An anonymous critic of the font Comic Sans didn't say that, for those are the words of its designer, Vincent Connare, talking to the Wall Street Journal. Connare followed up that comment, however, with this: "If you hate it, you really don't know much about typography, either, and you should get another hobby."

For a long time, something has been circulating on the Web, alongside all the double slashes and pop-up ads, to the effect of "There is great need for a sarcasm font." I have proposed that Comic Sans become the Official Sarcasm Font. This suggestion has gotten nowhere, perhaps because I keep proposing it in Comic Sans.

8. Tom Karen — Raleigh Chopper.

Before the BMX arrived on the scene in the late 1970s, if you wanted a bike that wasn't of the drop-handlebarred racing variety, Raleigh's Chopper (pictured up top) was one of the few options. Loved by millions for its comfortable saddle, laid-back seating position, and those huge Harley Davidson-esque handlebars, it was one of Raleigh's best-selling bikes in the 1970s.

Hell, I'm old and I don't remember these. Deprived childhood.

9. Kamran Loghman — Pepper spray.

Kamran Loghman worked for the FBI in the 1980s and helped turn pepper spray into weapons grade material. He also wrote the guide for police departments on how it should be used.

Which was then promptly ignored.

10. John Sylvan — Coffee capsules.

"I feel bad sometimes that I ever did it," he said a few years ago. "It's like a cigarette for coffee, a singleserve delivery mechanism for an addictive substance."

Millions of people are going to disagree with him, and probably worship him as a god. I mean, come on, as far as addictive substances go, caffeine is probably the least problematic.

So that's it for April. Yes, for those of you following along at home because you've been home for the past month or more and have lost track of time, it's the end of the month. Tomorrow is May Day, and nothing else is changing.

Except the format of these entries. For a little while.

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