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.

June 2, 2021 at 12:04am

#1011143

Survivor

This one's been hanging out in my queue for a long time, not being deleted. It's a miracle!

Why Earth’s History Appears So Miraculous
The strange, cosmic reason our evolutionary path will look ever luckier the longer we survive.

Because if the planet had been destroyed, we wouldn't be here to note it?

This is known as an observer selection effect, and the same sort of bias might apply not only to perforated planes, but to whole worlds as well.

"Might?" It's also a type of survivorship bias, and kin to the Anthropic Principle.

When I was a kid, lo these many aeons ago, it was common for us to ride around in the backs of cars or pickups without car seats, or even seat belts. I did it, and I'm still alive, so obviously from my point of view, that was a perfectly acceptable practice.

If, of course, you don't look at childhood death/injury statistics from that era.

It could be that we’ve been shielded from these existential threats by our very existence.

Rarely have I seen a more perfect reversal of cause and effect. It's remarkable, really.

As Sandberg and his co-authors Nick Bostrom and Milan Ćirković write, “The risks associated with catastrophes such as asteroidal/cometary impacts, supervolcanic episodes, and explosions of supernovas/gamma-ray bursts are based on their observed frequencies. As a result, the frequencies of catastrophes that destroy or are otherwise incompatible with the existence of observers are systematically underestimated.”

In other words -- as I interpret this, anyway -- we've been lucky so far, but might not be so lucky in the future. Based on the next few paragraphs in the article, I think I got it right.

If this is true, it might explain why our radio telescopes have reported only a stark silence from our cosmic neighborhood.

Or maybe it's because -- and yes, I know I've said this before -- what we're looking for is artificial signals, but evolution doesn't require the eventual emergence of technology-capable life. All it requires is fitness for survival, which takes many different forms; those qualities needed to build radios and rockets are irrelevant to species survival. Sure, one could argue that it allows for species survival through dispersal to other planets, but that's an effect, not a purpose.

One of the leading interpretations of this quantum weirdness is that all of the possible realities for the particle that were winnowed away in this act of observation actually are realized somewhere in branching-off parallel universes, by observers in parallel universes—parallel universes just as real as the one in which we happen to live. Though the universe may be infinite in distance it may also be infinitely divergent in this sort of ontological zoo.

Quibble: not actually infinite. But might as well be from our limited perspective, since the numbers involved are so freaking large.

This condition of eternal torment, where one might survive arbitrarily long by subjectively navigating the narrowing tributaries of the many-worlds time lines, staying alive through increasingly—and eventually astronomically—unlikely life paths, is known as quantum immortality, or quantum hell.

I comment on these things as I read them, not after (usually). I was just going to bring up the concept of quantum immortality. But I like "quantum hell" better.

“And eternal inflation implies an infinite universe. If there’s an infinite universe then yeah, if we wipe ourselves out there will still be plenty of life in the universe somewhere. And plenty of humans ... and plenty of Peters and plenty of Anthonys sitting in offices.”

This is a common misconception about infinity. The concept of infinity doesn't mean that there's definitely another "you" somewhere. Consider the set of positive integers, which is an infinite set. Never, not once in that entire set, does an number repeat. Sure, individual digits appear in different numbers, but there's only one 1, only one 42, only one 10¹⁰⁰. Infinity doesn't imply duplication.

Of course, the quote I quoted above was from someone far smarter than I am, so take my comment as you will.

To summarize, the article is worth contemplating, but I have to stress that it's all speculative. Philosophical, even. When I consider how many times I've come close to dying, but didn't, sometimes I wonder if the many-worlds hypothesis holds true, and the universe split into one containing alive-me and one containing dead-me.

But no, this is literally what Occam's Razor was invented for: "Never multiply entities unnecessarily." The many-worlds hypothesis may be compelling to physicists, but it's currently both unverifiable and unfalsifiable, and it literally, actually multiplies entities unnecessarily, when a more elegant answer is that we're only around to talk about this sort of thing because nothing has happened to prevent it.

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