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 15 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 17 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.

April 12, 2020 at 12:20am

#980839

A Certain Point of View

Today's article is a couple of years old now, but from what I've seen in the news, it's relevant to our current situation.

https://theconversation.com/the-thinking-error-at-the-root-of-science-denial-960...

The thinking error at the root of science denial

Currently, there are three important issues on which there is scientific consensus but controversy among laypeople: climate change, biological evolution and childhood vaccination.

Sigh. Four, now. I just saw something today that indicated there's a great deal of overlap between those who deny climate change and those who ignore social distancing guidelines.

This widespread rejection of scientific findings presents a perplexing puzzle to those of us who value an evidence-based approach to knowledge and policy.

I'm the first to admit that science isn't perfect -- it has a history of false starts, wrong turns, and outright reversals. The one I keep going back to is research on the benefits, or harm, associated with eggs in one's diet. In my lifetime, eggs have switched from good to bad and back again at least four times, probably more; I've lost count.

The problem isn't that science isn't perfect, though. The problem, apparently, is that people want something to be perfect and won't settle for less.

Spectrums are sometimes split in very asymmetric ways, with one-half of the binary much larger than the other. For example, perfectionists categorize their work as either perfect or unsatisfactory; good and very good outcomes are lumped together with poor ones in the unsatisfactory category.

In the early days of streaming Netflix, they had a 5-star system for rating their movies / shows, much like we do here on WDC. It allowed for nuance. Maybe there's something you like about a particular movie and something you dislike, so you gave it three stars. Some years ago, they switched to the very Roman thumbs system. Either the gladiator (or Gladiator, in the case of Netflix) lives, or he dies. There's no in-between, no room for mixed feelings.

At that point, I quit rating shows on Netflix. In fairness, I blame Siskel and Ebert from when there was a Siskel and Ebert. But that just goes to show: I don't think in binary; I can understand, sort of, why they did it, and still refuse to participate.

This sort of thing has bothered me for a long time. I've known people who really had this binary system in their own heads. To them, everything, to paraphrase the great sages Beavis and Butthead, either sucks, or it's cool. There's no in-between.

In my observations, I see science deniers engage in dichotomous thinking about truth claims. In evaluating the evidence for a hypothesis or theory, they divide the spectrum of possibilities into two unequal parts: perfect certainty and inconclusive controversy. Any bit of data that does not support a theory is misunderstood to mean that the formulation is fundamentally in doubt, regardless of the amount of supportive evidence.

Whereas, as I've noted before, I can never be 100% certain about anything. I give the sun a very close to 100% chance of rising tomorrow, but hey, you never know. Very close isn't absolute.

There is no ‘proof’ in science

In my view, science deniers misapply the concept of “proof.”

I can't stress this point enough. I agree, with a high level of certainty.

Deniers exploit the distinction between proof and compelling evidence by categorizing empirically well-supported ideas as “unproven.” Such statements are technically correct but extremely misleading, because there are no proven ideas in science, and evidence-based ideas are the best guides for action we have.

They take this even further, I think, into the realm of "what makes experts any better at this than I am?" In politics, perhaps, your opinion is just as valuable as anyone's; our whole political system is founded on that hypothesis. Science doesn't work that way. Someone who's studied, say, the inner workings of a cell and how a virus attacks it, is going to know more about covfefe-19 than someone who's studied auto repair.

I have observed deniers use a three-step strategy to mislead the scientifically unsophisticated. First, they cite areas of uncertainty or controversy, no matter how minor, within the body of research that invalidates their desired course of action. Second, they categorize the overall scientific status of that body of research as uncertain and controversial. Finally, deniers advocate proceeding as if the research did not exist.

And there is always uncertainty, as I've noted above. Additionally, knowledge gets refined over time. To pick a less controversial field, consider physics. Newton developed a theory of gravity that predicted, to a high degree of accuracy, the orbits of planets. Astronomers used this theory, combined with observations of the seventh planet (it shall remain nameless to forestall bad puns), to predict where the eighth planet should be, and behold, Neptune was discovered, right where the theory predicted.

And yet, it wasn't perfect. They found small discrepancies in the orbit of Mercury. It took Einstein to figure that out, and his equations refined - not overturned, but refined - Newton's. There is some evidence that they might need to refine Einstein's theories as well. And yet, when it comes to our everyday lives, and practical engineering of, say, bridges and buildings, Newton's equations are all you need.

I think it's a very human characteristic to want certainty. The danger lies in assuming that, since we can't have it, then everything is equally uncertain.

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