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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Show Recent Comments

10 Most Recent Blog Comments:

Turkey DrumStik

about 9 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

October 6, 2018 at 12:59am

#942781

Reality and Other Falsehoods

Brandiwyn🎶

: Science points to all kinds of things that we CANNOT sense as being real, proven indirectly by empirical data and math.

*points to black holes*

*points to the infinity of the universe*

So who's to say aliens / demons / monsters / zeitgeists (Zeitgeister?) / (insert the inimical presence of your choice) aren't real? Just because we don't yet have quantifiable evidence (direct or indirect) doesn't mean they don't exist. (As a reminder, Earth is round and circles the sun, and molecules are made of atoms are made of subatomic particles.)

I'm just trying to help you get over your nightmares by pointing out that they may be physically real, not just real in the sense that synapses are really firing. Just because we haven't proven it yet doesn't mean we won't eventually. So don't worry, you may not actually have a sleep disorder.

This is exactly what I'm talking about when I wrote: The more you try to define "reality" and "illusion" as they relate to each other, the more slippery it gets.

Incidentally, B, what you said doesn't frighten me in the slightest or change my worldview. You did a much better job of that with one picture of a Hello Kitty food truck. (shudder)

I also wrote: Can I sense it? Yes? Then it's real. It does not logically follow from this that if I can't sense it, it's not real. I'm pretty sure black holes exist because a) science predicted them and b) we can see their effects on other matter and infer that there is, in fact, a black hole there. It's like sitting nice and comfy inside your house and seeing, through the window, trees swaying - even if you can't hear the wind, the swaying trees are a dead giveaway.

But one must beware of taking even science at face value. Consider, especially, nutritional science. Over my lifetime, for example, eggs have gone from bad to good to bad to good to bad to... I think they're supposed to be good again? Or, after several studies that seem to indicate that moderate alcohol consumption is good for you, another study came out flatly against any amount of alcohol.

There are good reasons for this sort of thing - nutritional studies are, probably more than any other branch of science, subject to confirmation bias, problems with small sample sizes, and extraneous factors. To use the egg thing as an example, you can't exactly get 10,000 people together and have half of them eat nothing but eggs, and the other half eating nothing but egg placebos. There are other factors involved in any nutritional study results.

As for the alcohol thing, what that particular study failed to take into account was that if you take away one of the primary things that makes life worth living (booze), there's simply no point in living longer.

This sort of thing erodes trust in science and the scientific method, to everyone's detriment. So, at this point, we have a good half the population believing that climate change is a hoax, and a not-insignificant number of people insisting that the Earth is flat.

Science works. It's the best way we have of understanding the universe. It's not always right, and it's subject to changes and even reversals, but once something has been tested and observed enough, there's a high level of confidence in the result. There's a very high level of confidence that the Earth is roughly spherical. There's a very low level of confidence that drinking is always bad (and we need to be careful how we define "bad.")

Incidentally, the science is still split on the "infinity" of the universe. Could be finite but huge. Could go either way. What is known to a very high level of confidence is that the universe is bigger than we can possibly see. Infinity strikes me as a mathematical construct, though, not something with any sort of physical equivalent - but then we're getting into speculation and philosophy, not hard facts. I am certain, however, that given our short lifetimes, the cosmic speed limit, and the size of even the observable universe, it might as bloody well be infinite for all practical purposes. Hell, even if we could live a billion years, we couldn't closely observe more than a small fraction of the universe.

Anyway, if you want to get into things we can't directly observe but are pretty sure about, try dark matter and dark energy. Science shows that they make up the vast majority of the stuff in the universe. They're called "dark" specifically because we can't see them. We don't even know what they are, though there are guesses that are being tested right now - without much success. Might be it'll turn out to be something that turns all of physics on its head, the way Einstein's insights into the speed of light destroyed the idea of the "luminiferous ether."

At the same time, there are things that are claimed to exist that there's flimsy to no evidence for, or even evidence against: Bigfoot, space aliens, the Illuminati, ghosts, unicorns, and honest politicians, for example. That doesn't - and shouldn't - stop speculation about them, or imaginative writing. Some of these things might actually exist - especially space aliens (because see above discussion about the enormously, mind-boggling, effectively infinite, vast size of the universe) - but until we have real, hard evidence, it's speculation and guesswork.

That is not to say that these things aren't real. To wax Platonic (or Kabbalistic), anything you can think of has a reality of sorts. But again, my worldview is mostly practical, so I'll stick with what I can sense directly - or what science has shown to a high level of confidence has a material reality.

(The title of this blog entry is a reference to the title of a book found in The Elder Scrolls series of games. I find it amusing.)

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