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 10 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 12 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 13, 2020 at 12:55am

#985561

Shoot for the Moon

Entry #4 for "Journalistic Intentions"

[18+]

14. Shoot for the moon, and even if you miss, you'll get a lot of practical experience with Newtonian orbital mechanics.
Robert Waltz

I've been choosing these at random, and just happened to pick the one prompt that I twisted. And that's fine; it means I get to talk about two of my favorite things: motivational quotes and orbital mechanics!

I might even get a few words in about beer.

You certainly know the original quote: "Shoot for the moon. Even if you miss it you will land among the stars." For once, I could be arsed to look it up, and it's attributed to someone named Les Brown, who I've never heard of unless it was this kid I went to school with who, obviously, wouldn't be a kid anymore. Huffington Post, however, attributes it to Norman Vincent Peale, who I have heard of but couldn't name anything he wrote off the top of my head.

Yet another source says it was Neil Armstrong who said it, and if you haven't heard of him, boy howdy are you reading the wrong blog. Another claims someone named W. Clement Stone wrote a similar saying.

I think it's pretty safe to say that the actual author of the original quote is unknown, which is fine, because whoever said it ought to be fucking ashamed of themselves (I highly doubt it was Neil Armstrong). It's glurge of the lowest order, even worse than the vast majority of motivational quotes, and it contributes to the general misunderstanding of science that keeps people in willful, comfortable ignorance.

Why do I doubt it was Armstrong? Because he's one of the dozen or so people who have been to the goddamned moon, in the course of training for which he would have known exactly and in excruciating detail what would have happened if he'd missed: that they'd be in orbit indefinitely, around Earth, moon, or sun, and whether he and Aldrin and Collins would have died of asphyxiation, dehydration, hunger, or the collective miasma of each other's farts before their desiccated corpses might be found by some other team of astronauts from some other country in the distant future, because if that had happened, we'd have scrapped the space program forever and the only other country who might have sent people to recover them wouldn't have gotten there quickly enough to save them.

Hence my twisting of the quote.

Orbital dynamics is weird and counter-intuitive. To slow down, you have to speed up. To speed up, you have to slow down. There is no left, right, up, or down. There's no such thing as moving in a straight line as we understand it from our Earthly perspective.

Worst of all for the purposes of this idiotic quote, and I'm going to use miles here but as you'll see it doesn't matter, the moon is about 240,000 miles away off the top of my head. The Earth is about 21,000 miles in circumference, also off my noggin, so you'd have to circle the Equator (or any line of longitude, etc.) more than 13 times to travel an equivalent distance.

That's a long way, but it's a rounding error compared to the distance to even the nearest star.

The nearest star, Proxima Centauri, is about four and a quarter light-years away. Now I've reached the limits of being able to pull numbers out of my ass, so I'm going to look up the conversion. That's (grâce à Google) 2.5e+13 miles. 240,000 is 2.4e+5 in the same notation. The reason we use scientific notation is that such distances are absolutely unimaginable to most of us.

And that's only the nearest star.

If you take "among" literally in "among the stars," maybe you can accept 1/3 to 1/2 of the distance from our sun to Proxima Centauri, but even that is a mind-bogglingly huge distance. With current technology, you'd literally never get there; we just don't know how to go fast enough to go that far in one human lifetime, even ignoring such trivia as bringing along enough food, water, and beer to keep you alive on the journey.

The original quote is meant to convey the idea that one should dare to dream big, and even if one fails at one's goal, something even better might happen.

But in reality, you just die, quickly or slowly.

And that's reality, folks. But that doesn't mean there's not hope. There are two reasons why we, as a species, were able to put people on the moon to begin with: science and technology.

True, we've dreamed about it for a very long time, from our limited perspective of time. This

relates some of the historical tales about the journey.

One such story tells of the journey Lucian and 50 companions take on a boat carried to the Moon by a giant waterspout. When they arrive on the lunar surface, they’re greeted by a race of three-headed vultures and soon find themselves in the middle of a war with another species. Eventually they make their way back to Earth and experience more fantastic adventures. Lucian’s lunar tale is the earliest known piece of fiction that depicts space travel, a Moon landing, aliens, and interplanetary war.

Hell, that sounds like an awesome story. But a story is only the first step. A giant waterspout isn't going to carry you to the moon; it's more likely to rip you to shreds. Similarly, radiation isn't going to turn you into the Hulk, toxic waste isn't going to give you heightened senses, and getting bitten by a radioactive spider is a major nope. Stories are great; they make us human, connect us to each other and the world, and I daresay they're what most of us are here for. But it was Newton who ultimately showed us how to get to the moon, not with stories, not with motivational quotes, but with goddamned mathematics.

But hey, I hated the quote enough to give this much thought to it, so I guess it's useful after all.

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