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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Write_Mikey_Write!

about 6 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.

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Adherennium Plotting something

about 8 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.

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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').

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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, 2020 at 12:05am

#989758

Dueling Scientists

Today's article was published in the Before Times, but its theme is very relevant today.

https://theconversation.com/when-science-gets-ugly-the-story-of-philipp-lenard-a...

When science gets ugly – the story of Philipp Lenard and Albert Einstein

Never heard of Philipp Lenard? Neither had I until I read this article. His relative (pun intended) obscurity undermines the point that I think the article is trying to make. Let's take a look, shall we?

Scientists are not always as scientific as many suppose.

Well, duh. Scientists are humans. I know shows like to portray them as some sort of wizards, the Gandalfs of technology, but they're all people and thus subject to all of the biases, contradictions, prejudices, and just plain wrongness that plague the rest of us.

Recent well-publicized cases of scientific fraud prove that scientists can be as susceptible to the allures of wealth, power and fame as politicians...

Interesting comparison there. In science, theories that turn out to not fit observations are eventually discarded into the trash heap. In politics, well, for example, here in the US we have two houses of Congress and three branches of government, lots of different people thinking about stuff, so that eventually bad politics also gets discarded. Usually. Eventually.

Such breaches prove that scientists do not always base their work strictly on rigorous experimentation, data collection and analysis, and hypothesis testing.

"Prove" is kind of a dirty word here. I might have picked "demonstrate." That is, like I said, scientists are humans and thus fallible. However, that we know about these instances of bad science shows that science, as a process, is working as intended -- to smooth out the bumps in the road caused by individual bias and error.

In fact, scientists frequently disagree with one another, both as individuals and as representatives of competing schools of thought.

Feature. Not a bug.

The article goes on to provide a brief bio of both Lenard and Einstein. Lenard was "a German experimental physicist," while Einstein was "a Swiss theoretical physicist."

It's no secret that experimental and theoretical physicists are often at odds. They need each other, and collaborations have happened, but my impression is that, historically, members of each group consider the other somehow inferior. But, again, this is the intersection of humanity and science; eventually, the experimenters will either find results consistent with the theories, or they won't, in which case the theory has to be abandoned or modified.

Again, this is how science is supposed to work. It's not like scientists are supposed to be all one happy family, always agreeing with each other.

Lenard, meanwhile, was soon swept along in a wave of German nationalism that accompanied World War I. He became increasingly convinced of the existence of a distinctively German physics that needed to be defended against the plagiarized or frankly fabricated work emanating from other countries. Lenard also became more and more mired in anti-Semitism, accusing the “Jewish press” of, among other things, promoting Einstein’s dangerous work on relativity.

Ideally, science is universal. But scientists aren't always.

Lenard’s attacks on Einstein became increasingly vitriolic. He compared theoretical physicists to Cubist painters, who in his view were “unable to paint decently.” He lamented the fact that a “Jewish spirit” had come to rule over physics.

From everything I've heard, Einstein wasn't even a practicing Jew. So it wasn't even about religion; this was about ancestry.

Lenard’s conviction that science, “like everything else man produces,” was somehow grounded in bloodlines led him to become one of the early adherents of National Socialism.

And just to get this out of the way: Nazis were "socialists" the way North Korea is a "democratic republic." I can call myself a unicorn, but that doesn't make me a unicorn.

The story of Philipp Lenard reminds us that even scientists of the very highest caliber sometimes think, speak and act in utterly unscientific ways, swayed by prejudices that have no scientific basis.

And yet, as I said above, Einstein is a household word these days, while Philipp Lenard has been all but forgotten. Einstein's theories have been supported by evidence time and time again, from measurements of the precession of Mercury's orbit to the experiments in gravity wave detection. This is how it's supposed to work and, in general, it does.

And so science progresses. It's wrong to revere Einstein the man; I'm certain that he had his flaws just like everyone else. But again, his science has held up -- even though there seems to be growing evidence that they're going to have to tweak the equations to account for what they're calling dark matter and dark energy.

The problem comes in when people hear a pronouncement by someone with a degree or certification, take it as fact, and then ignore anything that contradicts it. So it's the last sentence of the article that concerns me:

They are human beings too, and members of the general public need to be careful to distinguish between a scientist whose arguments are based in evidence and one whose pronouncements stem from other, less reliable sources of conviction.

That's not always easy to do. Evidence, especially at the highest theoretical levels, isn't really accessible to "members of the general public." I mean, it is accessible, but not always comprehensible. But I know this much: if a so-called scientist, or doctor, or whatever, starts spouting off about demon sperm and lizard aliens, I think she can, and should, be summarily ignored -- at least until we have some hard evidence of lizard aliens.

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