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!

a day 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

a day 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.

October 25, 2019 at 12:15am

#968390

In Extremis

A rare "numbers" entry today.

http://nautil.us/blog/how-to-understand-extreme-numbers

How to Understand Extreme Numbers

One of the many things that bug me but probably shouldn't is when someone uses the expression "almost infinite." This grates like "very unique," which is different in that it's an unnecessary intensifier, while "almost infinite" is utterly meaningless.

There are, of course, things that might as well be infinite for all that we're ever going to be able to count them individually, such as the number of atoms in the observable universe. A huge, immense, unbelievably large number in our experience - but just as far away from infinity as the number 42.

You know what else bugs me a lot? Pedantry. So I'll shut up about this now.

Not a lot of research has been done on how our minds perceive and comprehend large orders of magnitude—big differences between the size of, say, a cell and our sun.

One of the things I appreciate most about science communication is when someone gives relative sizes, such as "an atom is as much smaller than a human as a human is to the visible universe." (Note: I didn't look this up, so I may be a bit off on the actual relative sizes; don't quote me on this.) Such comparisons make things that are very hard to visualize slightly less hard to visualize.

But about 35 percent of people in the study used what the authors call a “segmented linear heuristic.” That means they correctly distinguish between numbers within the millions or billions, but assume that “million,” “billion,” and “trillion” are equally spaced on a number line. They were generally great at comparing the relative sizes of numbers like 2 million and 800 million, but many treated 980 million and 2 billion as nearly identical.

Well, we're used to logarithmic scales on big-number charts, so that doesn't surprise me much.

Of course, our ancient human ancestors didn’t live among billions of people, or incur trillions of dollars of debt. The orders of magnitude in their immediate surroundings were limited to what they could experience firsthand. It’s not surprising that we can intuitively visualize a 6-inch or 6-mile distance, but 93 million miles to the sun seems…super far.

Ah, yes, the inevitable and unsupported-by-evidence "evolutionary psychology" nod. I wish they'd stop that shit. (I'm not saying they're wrong; just that it's basically guesswork.)

So how can we make large numbers more easily graspable? A group at Microsoft is working on it.

And then you, too, can grasp the concepts of large numbers, for a low subscription price of $79.99/year. If you ever forget how to do it, just turn your brain off and then on again.

When scientists navigate their way through extreme numbers in their daily work, they aren’t constantly comparing enormous or miniscule measurements to units of everyday life. Instead, their fields have their own perspectives relative to different units.

You know what I learned today? Well, technically yesterday. You know what I learned yesterday? I learned that quantum physicists have a name for the amount of time it takes for a photon to cross the distance of the width of a proton. I shouldn't have to tell anyone this, but that is a mind-bogglingly small amount of time. So instead of saying, like, 10^-24 seconds (or whatever the actual value is) they call it... a jiffy.

I'm not kidding. https://en.wikipedia.org/wiki/Jiffy_(time) (Other disciplines use the word for other measures of time, apparently.)

Same kind of reason why cosmologists measure distances in parsecs or light-years instead of kilometers, only in the other direction.

“Things that are so far removed from our daily experience—like quarks, and dinosaurs, and Kim Kardashian—are inherently hard to understand,” extreme numbers included.

Oh, look, a funny guy mathematician.

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