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 3, 2021 at 12:02am

#1014973

Tales from the Cryptid

Confession: I kinda like cryptids.

Why We Can’t Rule Out Bigfoot
How the null hypothesis keeps the hairy hominid alive.

It's not that I "believe in" Bigfoot (or Nessie, or the Jersey Devil, or chupacabra, or whatever). It's that I'm fascinated by what the stories say about us and about the way we approach knowledge.

This article is almost exactly seven years old now, but it's probably not like the science has advanced...

The international collaboration of scientists, led by University of Oxford geneticist Bryan Sykes, found no evidence that the DNA from the hairs belonged to a mysterious primate. Instead, for the most part, it belonged to decidedly unmysterious mammals such as porcupines, raccoons, and cows.

All animals (well, all things, really) are mysterious until they're not, and then a lot of them keep on being mysterious even as we study them. For instance, the opossum. If I didn't have a few climbing up on my deck to say hi every once in a while, I might not believe they existed. Hell, the first report of the Virginia opossum comes from John Smith

(yes, that John Smith, not one of his many namesakes). From that link: Strachey's notes describe the opossum as a "beast in bigness of a pig and in taste alike," while Smith recorded it "hath an head like a swine ... tail like a rat ... of the bigness of a cat." Okay, well, I think Strachey was hitting the hooch there; Smith's notes are a bit closer to the opossums I've seen.

Anyway, Bigfoot.

People often think that the job of scientists is to prove a hypothesis is true—the existence of electrons, for example, or the ability of a drug to cure cancer. But very often, scientists do the reverse: They set out to disprove a hypothesis.

Which is what I've been saying.

Bigfoot advocates have repeatedly claimed that professional scientists are willfully ignoring compelling evidence. The problem, in fact, is that the advocates haven’t been approaching the question of Bigfoot in a scientific fashion. So two years ago Sykes and his colleagues decided to run a scientific study of those hairs from an “anomalous primate.” And that involved creating a null hypothesis to try to reject.

Unlike some people, I don't have a problem with science being science even when the hypothesis is, at first glance, absurd, as in the case of most cryptids. At the very least, negative results provide skeptics with more argumentative ammunition. I'm not what you'd call a hard skeptic: show me compelling evidence and I'll change my tune. This applies to cryptids as well as psychic abilities, faeries, space aliens or whatever.

Thing is, everyone's level of "compelling" may be different.

The results were clear: The scientists found precise matches for all 30 samples in previously known mammals.

Does this mean Sykes and his colleagues have proved that Bigfoot does not exist? No... The question remains open, and—if Bigfoot doesn’t exist—always will.

Which is a long way to go about showing once again that it's very, very difficult, if not impossible, to prove a negative. Russel's Teapot

comes to mind.

That’s not to say Sykes’ study didn’t offer its own surprises. Two hair samples from the Himalayas matched a DNA sequence that was extracted from a 40,000-year-old fossil of a polar bear. Stranger still, their DNA was not a match to living polar bears.

And this is why I like to keep an open mind about these things. Even if we never find Bigfoot (and given that the classic Bigfoot sightings were revealed to be guys in ape suits) (well, hairy ape suits; we already wear ape suits), who knows what strange and cool things we might discover along the way?

Besides, it makes excellent story fodder.

Just to be complete, I'll save you a search: here's the Wiki article on Bigfoot.

6 Like • Comment • View 1 comment

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