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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 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 10 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.

March 30, 2024 at 11:07am

#1067184

Dis or Dat Information

What I'm sharing today, from Men's Health, is two and a half years old, so its main target is what was plaguing everyone's lives in October of 2021. Nevertheless, it's still relevant.

The Golden Age of Junk Science Is Killing Us
Misinformation is being spewed, weaponized, and consumed at a deadly rate. Fortunately, there's a way out. Here's how to make sense of what you're seeing.

Considering that not much has changed on the misinformation front, we didn't take the "way out."

In a recent Economist/YouGov poll, 20 percent of U. S. citizens surveyed said they believe that Covid vaccines contain a microchip.

Again, "recent" here is 2021. Obviously, 20 percent is about 20 percent too high, even if you account for poll trolls. But it's a lower number than I, a pessimist, expected.

This survey also found that only 46 percent of Americans were willing to say that the microchip thing is definitely false.

Okay, that's more in line with what I expected.

It is becoming harder and harder to tease out the real from the unreal. Sense from nonsense. Magical thinking from microchips.

Yep. Just look at the nutters trying to attribute the Baltimore bridge disaster to, well, pretty much everything except mechanical failure.

Part of the problem is that we have normalized nonsense in some very subtle and some very obvious ways. Heck, there are a host of (very) successful wellness gurus who have embraced pseudoscience as a core brand strategy.

"A host of?" No, it's all of them. Unless your wellness "guru" says something like "follow evidence-based medicine," which none of them do.

We really should come up with a better term. Using guru like that has got to be insulting to Hindus and Buddhists.

And thanks to people like Andrew Wakefield—the disgraced former physician who started the vile “vaccines cause autism” fallacy in a paper published in and later retracted by The Lancet—misinformation about vaccine safety has continued to spread and find new audiences.

As I've noted before, the fallacy got lots of attention. The retraction did not. People still believe that bullshit.

Besides, even if it were true, which it's not, a person's fear in that area reveals a great deal about how they really feel about the neurodivergent.

But there is a way forward! By using a few critical-thinking tools and being aware of the tactics used to push misinformation, we can cut through the noise.

Yeah, but we didn't.

We know that misinformation can spread fast and far. In August, Facebook released a report on its most widely viewed content from January through March 2021. The winner? The post seen more times than any other was a misleading article implying the Covid vaccine had killed someone. This nugget of misinformation was viewed nearly 54 million times by Facebook users in the U. S. in that three-month period and has been leveraged by countless anti-vaccine advocates, compounding its impact.

Even if the vaccine (or any course of treatment) did cause a death, what we have to do is weigh that against the expected number of deaths from doing nothing.

It's pretty well-known, for example, that, on occasion, people die during surgery. Maybe from an unexpected reaction to anesthesia, or whatever. Would that mean we should stop dong medically necessary surgery, which many more people would die from the lack of?

People make fun of the philosophical Trolley Problem, but there's a practical example of it right there.

And that's not even getting into the fact that people die every day (in general, I mean; obviously, no one person dies every day), so it's inevitable, statistically, that some of them will die of unrelated causes after medical treatment. It's like "Frank ate a turkey dinner, then dropped dead! No way am I ever again eating turkey! Or dinner!"

Another problem: Lies, fake news, and pseudoscience can be made more compelling (microchips in the vaccines!) than the boring old truth (safe, clinical-trial-tested, actual vaccine ingredients). Indeed, research has found that, yep, as the saying goes, “a lie can travel halfway around the world while the truth is putting on its shoes.”

Fact-checking a statement takes many orders of magnitude longer than making the statement. And, because of primacy bias, often, fact-checking does absolutely no good whatsoever.

The Wakefield disaster is a perfect example of primacy bias. "Vaccines cause autism!" has way more staying power than "No, they dont," even though the first statement is provably false.

The article also has something to say about my favorite bias, confirmation bias.

So, I do what I can.

It's not enough.

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