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.

October 7, 2019 at 12:18am

#967365

It Never Gets Old

https://www.vox.com/2019/8/8/20758813/secrets-ultra-elderly-supercentenarians-fr...

This flat-out amuses me.

Study: many of the “oldest” people in the world may not be as old as we think
A new paper explores what “supercentenarians” have in common. Turns out it’s bad record-keeping.

Every now and then, some news source runs out of actual news and does a fluff piece about the "oldest person in x" where x is a state, a country, or the universe. This is a massive jinx, because shortly afterward, the second-oldest person in x moves up a rank.

But you know what would suck worse than people shoving cameras and microphones in your face when you're like, 110 years old? Being #2 and watching #1 getting all the media attention, and thinking, "Oh shit, I'm next. If I'm lucky. Let's see... how did I reach my ripe old age? I know - I'll tell 'em 'drinking, smoking and fucking.'"

Old people make the best trolls because they no longer give a damn.

But it turns out some of them are flat-out lying. Or, to be charitable, have shitty memories. That happens when you get superannuated.

We’ve long been obsessed with the super-elderly. How do some people make it to 100 or even 110 years old? Why do some regions — say, Sardinia, Italy, or Okinawa, Japan —produce dozens of these “supercentenarians” while other regions produce none? Is it genetics? Diet? Environmental factors? Long walks at dawn?

Bong hits.

A new working paper released on bioRxiv, the open access site for prepublication biology papers, appears to have cleared up the mystery once and for all: It’s none of the above.

Instead, it looks like the majority of the supercentenarians (people who’ve reached the age of 110) in the United States are engaged in — intentional or unintentional — exaggeration.

And that's what amuses me.

It seems unlikely that living in high-crime, low-life-expectancy areas is the thing that makes it likeliest to reach age 110. It seems likelier, the paper concludes, that many — perhaps even most — of the people claiming to reach age 110 are engaged in fraud or at least exaggeration.

"How'd you make it to 110?" "Identity theft.Burger King every day."

In other words, all of our research into the biomarkers, habits, and diets that predict extreme old age? Probably worthless, because a significant share of the sample was not actually as old as we thought.

Science is a powerful tool, but it's only as good as its inputs.

1 Like • Comment • View 4 comments

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