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 2 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 4 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').

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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.

December 24, 2022 at 6:58am

#1042100

Reason for the Sneezin'

There are few certainties in life, but two of them are: 1) you will get a cold and 2) you will get copious advice (most of it wrong) on how to deal with that cold.

Fact or Fiction?: Feed a Cold, Starve a Fever
The answer is simmering in a bowl of chicken soup

Turns out, though, that when it comes to chicken soup, my people might have been right all along.

Maxims typically date back many years, but “feed a cold, starve a fever” may beat them all. This saying has been traced to a 1574 dictionary by John Withals, which noted that “fasting is a great remedy of fever.” The belief is that eating food may help the body generate warmth during a “cold” and that avoiding food may help it cool down when overheated.

And balance the humours.

But recent medical science says the old saw is wrong.

I'm shocked—SHOCKED—that 16th century medicine turned out to be wrong.

Of course, not all of ancient medicine was wrong. It just needs to be examined scientifically.

Let’s take colds first. When your body fights an illness it needs energy, so eating healthy food is helpful.

For various definitions of "healthy."

The reasons to eat for fever are more interesting. Fever is part of the immune system’s attempt to beat the bugs. It raises body temperature, which increases metabolism and results in more calories burned; for each degree of temperature rise, the energy demand increases further. So taking in calories becomes important.

This is the refutation of the "starve a fever" part.

Even more crucial is drinking.

One time I was sick and the doctor insisted I drink "plenty of fluids." I pointed out that tequila is a fluid. He was not amused.

“You have to make yourself drink fluids, even though all you want to do is collapse,” says William Schaffner...

It's not that I want to collapse, usually. It's that colds and flu make everything taste bad to me. Water especially all tastes like it just came fresh from the sewer. I've been lucky (actually skilled and careful) so far and not had covfefe-19; the most terrifying symptom of that disease is loss of smell and taste (death would be preferable), but I'd almost rather not taste anything than have familiar delicious beverages taste like horse piss.

Given the wisdom noted above, Schaffner says, don’t force yourself to eat if you don't feel like it. “But drink,” he adds. “It’s the liquids that are important.” Avoid caffeine and alcohol. Caffeine enhances dehydration. So does alcohol, and it is also a depressant, holding us down.

That leaves nothing for me to drink. Piss off with that.

While I'm mostly joking about the alcohol, caffeine is the only thing that gets me through sneezin' season. I despise coffee, though, so I get lower amounts in things like tea and Coke.

What about some other common conceptions for beating colds and fevers, such as eating chicken soup?

One thing I've never been real clear on is whether one "eats" soup or "drinks" it.

Chicken soup doesn’t possess any magic ingredients, but it has calories as well as the all-important liquids again. The warm vapor rising from the bowl can also moisten and loosen dried mucus.

Ew, but true. So yes, chicken soup does help with cold symptoms. Other hot beverages (or liquid foods) could also. I suspect that the placebo effect is in play for many people; they believe chicken soup will help, so it does.

And like the article implies, it can't hurt. Unlike some of the advice you get heaped upon you when you have a cold.

Supplements are dubious at best. The data from studies about taking vitamin C are inconclusive, as they are for zinc.

Pretty sure vitamin C is relatively harmless. Zinc, however, sounds like something you can overdose on. Fun history fact: when I was in college lo these many years ago, I was a guinea pig in a common cold zinc study. As it has now been nearly 40 years since that study, and zinc use is still inconclusive, I'd have to conclude that zinc for common colds is bullshit.

Solid studies of echinacea show no benefit.

There's one benefit, and it's to the bank accounts of echinacea supplement producers.

Over-the-counter remedies may or may not help, but that’s a whole ‘nother story. They can relieve symptoms but they do not kill off viruses or bacteria.

I have never, not once, seen any improvement in symptoms from taking antihistamines and/or decongestants. I wouldn't go so far as to say they're useless; they seem to help other people. But they have no observable effect on me.

One thing I'm a bit surprised about is that the article doesn't emphasize the uselessness of antibiotics treatment for viral infections. Not only is it useless, but it can be actively harmful in several ways.

Cold and fever germs usually run their course, and the immune system eventually gets the upper hand. In the meantime, drink drink drink.

That always sounds like a good idea to me.

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