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 an hour 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 3 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.

May 13, 2023 at 8:22am

#1049551

True or False: Nutrition Edition

From the "don't believe everything you hear" department (courtesy of The Guardian):

Chocolate doesn’t cause acne – but carrots do help you see in the dark: the best and worst health myths and wisdom
True or false: cheese gives you bad dreams and oysters are aphrodisiacs? We investigate good, bad and mad health advice

Folk "wisdom" usually isn't wisdom, but mythology. People have always had a problem confusing correlation with causation.

Sometimes, though, like a sightless person throwing darts randomly and hitting a bullseye, it turns out to be right—at least provisionally.

How do you tell the difference? Science, of course.

I won't copy all of them here; there are quite a few. Just hitting some highlights that I wanted to comment on.

Chicken soup helps cure colds and flu

Works best if prepared by a Jewish mother.

Okay, no, that's a joke. But I'm pretty sure the canned kind is going to be inferior to the homemade variety. I'm wary of the word "cure" in the title; however, this falls into the "can't hurt and might help" category. Unless you're vegan, in which case, good luck.

Anyway, I've banged on about chicken soup in here before. The short version is, if it makes you feel better, and you like it, great.

Chocolate causes acne

This one's labeled "false." As with much of "nutrition science," the jury's still out.

An apple a day keeps the doctor away

Also labeled "false," but only on a technicality: you're going to get sick eventually, no matter what you do or don't do. But, as the article notes, it's not going to hurt you to eat a damn apple. And it's admittedly a catchy rhyme.

Going out with wet hair gives you a cold

"False." Duh. Colds are caused by viruses. Viruses that you pick up from *shudder* people.

Carrots help you to see in the dark

The article points to "true," but I have to say, not to the extent that mythology would indicate. This nonsense started, if I remember correctly, in England during WWII, when, not wanting "zee Germans" to know about the Allies' sophisticated (for the time) radar, they attributed early warning of aerial attacks to people eating carrots and therefore seeing the impending threat better in the dark.

But again, as with apples, it's not like eating a few carrots is going to hurt.

Cracking your knuckles will give you arthritis

This one's false, and I've known that for some time, but goddamn, it's annoying. So if you use it to scare kids into not cracking their knuckles, I can understand that.

I crack my knuckles all the time.

It takes up to seven years to digest swallowed chewing gum

Another false one intended to scare children straight.

Garlic under your pillow aids sleep

Labeled "false," but I'd call it true, if you're frightened of vampires; clearly, you're going to sleep easier if you know you're protected. It also keeps other people away because of the smell, so you get a better night's sleep.

Still, when it comes to garlic, I'm too busy eating it to put it under my pillow.

Urine relieves jellyfish stings

I'm including this one in my commentary because I'm still hearing this nonsense sometimes. I suspect it got started by someone with a pee fetish, which is way more common than I ever realized.

Oh, yeah, and it's false.

Cheese gives you bad dreams

I remember the first time I heard of this one. It was in Dickens' A Christmas Carol, as Scrooge attributed his nocturnal visitations to possibly having eaten cheese.

As the article notes, dairy products might actually help with sleep. Again, good luck, vegans.

Probiotics support your gut health

This is the last one on the list, and it doesn't surprise me in the yeast—er, I mean, least—that it's not entirely true. The magical benefits of probiotics are mostly marketing gimmicks.

Big surprise.

7 Like • Comment • View 4 comments

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