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:

Write_Mikey_Write!

about 9 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 11 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.

February 24, 2024 at 10:04am

#1064779

Nerdy Words

Everyone knows that we nerds are generally immune to problems affecting normal people, such as STDs, sunburn, and athletic injuries. To make up for it, we have today's article, from Cracked:

5 Diseases That Only Hit Nerds

You may, however, come down with a whole series of other specialized conditions that will savage your body or will break your mind.

And no, one of them isn't "brain overheats from exertion."

Disclaimer: everyone is a nerd about something. This article is about the classic nerds who follow intellectual pursuits far more than is socially acceptable. Like me, for instance.

5. Nobel Disease

This isn't the same thing as noble rot.

When you get a Nobel Prize, the world is telling you you’re one of its smartest people. You may be set for life. So, there’s always the possibility of the recognition going a little to your head. Winners might go on to pursue ideas unconnected with their specialty, sometimes devolving into total nonsense.

First of all, this doesn't apply to the Peace Prize, which hasn't been relevant for decades. Second, I don't think anyone I know is in danger of contracting this dread malady. And finally, this just goes to show that even geniuses aren't immune from the Dunning-Kruger Effect. Which may not actually be a thing, but I say it is because I know better than They do.

4. Laptop Thigh

If your skin spends lots of time next to a heat source, you may come down with a condition called Erythema ab igne. That’s Latin for “redness from fire.”

One way English is superior to Latin is that we use single-syllable nouns for the most common things, such as cat, heat, and red. And nerd.

Many gentleman nerds already now the dangers of keeping laptops on their actual laps (it fries the testicles), but laptop thigh can affect anyone.

There aren't too many advantages to being a short guy. You're locked out of the dating pool—wait, no, that's an advantage. Another advantage is you don't have a lap, so "laptop" isn't the right descriptive. It does merge two simple one-syllable words, but, for instance, mine is always on a desk or table when I use it. Still, better than the French version: ordinateur portable.

3. Formaldehyde Hunger

People tend to become hungry in the close vicinity of corpses. This is dubbed “formaldehyde hunger,” on the assumption that the preserving chemical formaldehyde gets into people’s systems and stimulates their appetites.

I thought they mostly quit using formaldehyde, switching to a less carcinogenic preservative. But what do I know?

2. Brain Fag Syndrome

No, this isn't about a common slur for nerds, gays, and gay nerds.

The British diagnosed this syndrome in their subjects in Africa, who continued to use the term into the 20th century. Over in America, though, people doing lots of brain work were also experiencing mental fatigue. Some doctors dubbed this an exceptionally American problem, naming it “Americanitis.”

Nowadays, I'm pretty sure "Americanitis" is used for an inordinate love of firearms, eagles, and eagles bearing firearms.

The best treatment, they note, is rest. Yes, you feel better when you take a rest from work. These past 150 years of medical research have produced some marvelous breakthroughs.

This is as close as the list gets to "brain overheats from exertion."

1. Dysrationalia

English words for uncommon things are allowed to be multisyllabic, and derived from Latin and/or Greek.

When we’re measuring brain power, you’ve got your computational power, but then you’ve also got your ability to be rational, and this consists of a bunch of different types of intelligence. There’s reflective cognition. There’s epistemic rationality, your ability to make correct decisions free of various fallacies. There’s syllogistic reasoning, which requires discarding biases. Put it all together, and we find that some people with high I.Q.s score worryingly low in rationality. We describe such people as suffering from dysrationalia.

In other words, you can be really smart and still be really stupid. As anyone who's read my blog can attest.

5 Like • Comment • View 2 comments

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