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 2 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 4 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.

August 7, 2021 at 12:02am

#1015251

In the Past

Sometimes I just gotta go with a Cracked article.

55 Strange Facts About Famous Events From History

Since it's Cracked, I wouldn't put too much stock in the absolute veracity of all of these "facts." I mean, they don't lie on purpose like some websites, but they can't be arsed to do thorough fact-checking any more than I can.

Still, it's worth reading the whole list. I'm only going to comment on some highlights.

1. The French Revolution

Louis XVI and Marie Antoinette fled the palace and tried to go incognito when revolution broke out. But a postman thought he recognized them, and he confirmed his suspicions by comparing the king's face to the one on banknotes.

Nowadays, we have facial recognition software, and you don't have to be a monarch to get recognized by The System. Yet another reason to wear a mask.

7. The Challenger Explosion

Following the explosion, teams spent nine weeks searching a wide area for bits of wreckage. In the process, they stumbled on a duffle bag with 25 kilos of cocaine.

I should point out, because the article doesn't clarify this, that the nose candy probably didn't come from the shuttle, but just happened to be in the wrong place at the wrong time.

15. The Entente Cordiale

In 1904, England and France signed an agreement that ended almost a thousand years of conflict. They did so because King Edward VII wanted to visit French brothels.

I can think of worse reasons to sign a peace treaty.

19. The Hatfields-McCoy Feud

Records are vague about how this famous family feud started, but it now looks like the McCoys suffered from von Hippel-Lindau disease. This genetic condition creates tumors that cause "hair-trigger rage and violent outbursts."

If only they'd had a doctor in the family.

36. North African Campaign

Both sides during World War II left landmines in the African desert -- tens of millions of landmines. The climate preserved them so well that in the 21st century, ISIS dug up minefields as sources for explosives.

I want to know how they decided who was on mine-digging detail.

45. The Age Of The Samurai

High-ranking samurai strapped giant balloons to their backs. They inflated this bag, called a horo, to protect them from arrows.

I have Questions.

54. The Gold Rush

Gold wasn't the only way to make money when settlers headed to California. In 1851 and 1852, the state paid over a million dollars in bounties for killing or maiming Native Americans. One ad offered "$25 for a male body part, whether it was a scalp, a hand, or the whole body; and then $5 for a child or a woman."

I can see how that could easily be abused. I mean, of course it's abuse to start with, but I can see the bounty hunter now: "Here's a scalp, two hands and a body." "But the body is missing both hands and a scalp." "Huh. Go figure. I'll take my $100 in small bills, thanks."

Like I said, just highlights (or as in the case of that last one, lowlights) -- and we could use a little more historical perspective right about now.

5 Like • Comment • View 3 comments

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