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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Adherennium Plotting something

about 2 hours ago
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For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

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Brandiwyn🎶

Nov 12, 2024 at 10:50am
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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.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
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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.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
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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.

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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).

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Nov 7, 2024 at 10:58am
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Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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May 20, 2024 at 10:50am

#1071425

More Science

Speaking of science (yesterday's entry), today, the random numbers landed on this Big Think piece on the perception of science:

4 pervasive myths that cause us to abandon science
It’s not a gambit. It’s not fraud. It’s not driven by opinion, prejudice, or bias. It’s not unchallengeable. And it’s more than facts alone.

It's safe to say that I have a few issues with the wording in some places here.

When you think about what science actually is, how do you conceive of it?

I always picture Doc Brown's workshop in Back to the Future. Yes, of course I know it's wrong, but I'd rather be amused than right.

Do you do what most people do, and default to what you learned in our classes in school, with a layer of cynicism and skepticism layered atop it?

Hey! Cynicism and skepticism are my friends!

That’s understandable, as many of us remember being taught “facts” in our science classes that later were shown to be not as they seemed at the time.

This was, of course, not limited to science classes.

It’s as though, somewhere along the lines, many of us were taught isolated facts about the world, and our ability to regurgitate those facts was used as a metric to measure how good we were at science.

I've said this before, but memorizing trivia is an entirely different skill set from doing science. I'm not knocking trivia here; there's something to be said for remembering stuff without having to pull out your phone and google it. Plus, it's useful at bars sometimes.

Many of those facts may have felt absurd; many of the experiments we performed didn’t give the results we were told they should give; many of our experiences didn’t line up with what was written in our textbooks.

That's partly because some facts are, compared to our everyday experience, absurd. I mean, come on, the flow of time changes depending on how fast or slow you're going? That defies common sense! And yet it's demonstrably true; hence why I distrust "common sense."

If that’s what our experience with science is, then why should we be expected to believe that whatever science “tells us” is actually true?

Science isn't edicts handed down from on high. You're thinking of religion, or maybe law.

The way out, perhaps, is to slay the four most common myths that agenda-driven advocates leverage to sow doubt about bona fide scientific knowledge.

Science, in actuality, is both a process and the full body of knowledge that we have, and it adds up to the best, most truthful understanding we have of the world at any given time.

I'd put more emphasis on the "process" part, myself.

Here are four common lines of thought that we often make to argue against scientific findings, and why each of them is fundamentally flawed.

The article proceeds to do just that. It's not exactly short, and I'm not going to quote a whole lot.

(Myth) 1.) Science is biased by who funds it.

Because there are numerous instances throughout history where various industries have published junk science that does indeed promote their agenda — for example, the tobacco industry has famously been caught manipulating research — many people believe that this translates into all science being untrustworthy, particularly wherever it’s funded by some entity they view as ethically questionable.

One such instance that I remember referencing in here is how the "chocolate is good for you" result came from studies funded by Willy Wonka and performed by noted Oompa Loompa scientists who were paid in chocolate. It happens. (Okay, I'm being funny, but you get the point.) They usually get caught. The point here is that it's not right to generalize that to all science.

But part of valid skepticism is to look at the motivations behind research.

To be sure, there really is fraudulent research that gets conducted all the time; it’s one of the main reasons papers either get retracted or wind up being published in unscrupulous, sham journals.

One of the worst things about that is that people will remember the fraudulent science, and ignore or be unaware of the retraction.

(Myth) 2.) Science is driven by public opinion.

It’s often been said that, “Science doesn’t care what you believe,” but the truth is that science doesn’t care what any humans believe, in general. The Universe, as best as we can measure it, obeys certain rules and yields certain outcomes when we test it under controlled experimental conditions. The reason scientists so often find themselves surprised is that you cannot know the outcome of a new experiment until you perform it. The results of these experiments and the knowledge that comes along with it is available to anyone who reproduces the experiment. The results are found in nature, and anyone’s opinion on a scientific matter that’s been decided by the evidence is moot.

I'm just leaving this here because it's a pretty good summary of stuff I've been saying.

I just have one major quibble. Well, one major and one minor.

Major: saying that "anyone’s opinion on a scientific matter that’s been decided by the evidence is moot" is a bit misleading, in my opinion. For example, the chocolate nonsense I referenced above: you can have an opinion on it, but that opinion needs to be based on philosophy, not science. Like, I know the study was more than questionable, but I'll eat dark chocolate anyway, simply because it's delicious and I'm hedonistic.

Minor: there are some interpretations of quantum physics that point to the idea that the observer, in this case us, does influence the outcome of an experiment. What's in question is the validity of those interpretations, and how much they affect macroscopic phenomena.

I'm not quoting the other two points; they're also at the link.

In conclusion, for me anyway, while results are often provisional, science is the best method we've come up with for approaching legitimate knowledge. It's kind of like... maps. I've linked Atlas Obscura many times in here; hopefully, through that or by other means, you've seen historical maps. What was unknown on a map produced in, say, the 16th century, is generally presented as a best guess, an incomplete shoreline, a blob, a dragon representing the unknown, or whatever. As exploration improved, maps started to be drawn with greater and greater accuracy. Now, thanks to science, we have satellites doing our cartography with high precision.

No map, however, can have perfect accuracy. Zoom out too far, and small, but important, features disappear. In too close, and you lose the big picture. Rivers and oceans have variable water levels, so shorelines are, at best, averages. The Earth changes, and geological features (usually slowly) move: rivers change course, continental plates shift, erosion affects shorelines, etc.

This is an analogy and, like most analogies, it's not a perfect one. But if I'm going to try to circumnavigate the planet (which, by the way, is definitely round), I'll trust Google Maps before Magellan's sketches.

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