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 5 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 7 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.

July 23, 2022 at 12:01am

#1035578

No Way Home

A little bit more serious than usual today. Just the way the dice fall.

Bussed out
How America moves its homeless

The article is from way back in the Before Time, December of 2017. Something might have changed since then, what with travel restrictions for the pandemic. But it's still a good read.

Quinn Raber arrived at a San Francisco bus station lugging a canvas bag containing all of his belongings: jeans, socks, underwear, pajamas.

Pajamas? Luxury!

Cities have been offering homeless people free bus tickets to relocate elsewhere for at least three decades. In recent years, homeless relocation programs have become more common, sprouting up in new cities across the country and costing the public millions of dollars.

If only there were another way to use those millions to help the homeless. Gosh. I don't know. The answer eludes me.

But until now there has never been a systematic, nationwide assessment of the consequences. Where are these people being moved to? What impact are these programs having on the cities that send and the cities that receive them? And what happens to these homeless people after they reach their destination?

My guesses:

1) Anywhere else;
2) They're probably swapping so it's a wash;
3) They're still homeless.

In an 18-month investigation, the Guardian has conducted the first detailed analysis of America’s homeless relocation programs, compiling a database of around 34,240 journeys and analyzing their effect on cities and people.

That's a lot of work. Glad I didn't do it.

Some of these journeys provide a route out of homelessness, and many recipients of free tickets said they are grateful for the opportunity for a fresh start.

Well. Again, it pays to be pessimistic. That's pleasantly surprising.

While the stated goal of San Francisco’s Homeward Bound and similar programs is helping people, the schemes also serve the interests of cities, which view free bus tickets as a cheap and effective way of cutting their homeless populations.

I haven't been to SF since the Before Time, but spoiler: it's not working (sort of; you'll see later in the article). Also the font from the linked story is pretty small on my screen, so at first I thought that word was "culling."

Jeff Weinberger, co-founder of the Florida Homelessness Action Coalition, a not-for-profit that operates in a state with four bus programs, said the schemes are a “smoke-and-mirrors ruse tantamount to shifting around the deck chairs on the Titanic rather than reducing homelessness”.

You know, the "rearranging deck chairs on the Titanic" phrase is clichéd, but dammit, it's descriptive.

I'm skipping a bunch here. The good news is there's more nuance than I expected. The bad news is it's still not a comprehensive solution.

The interviews the Guardian has conducted with recipients of bus tickets indicate the outcome of their journeys can vary hugely.

Yes, better than I expected, but it sounds like some of the outcomes were even worse:

Last year Fort Lauderdale sent Fran Luciano, 49, back to her native New York to stay with her ex-husband, according to program records.

Lots of people get along with their exes. I do. But living with one? Fuck that.

But the money spent on bus tickets does not necessarily address the root causes of homelessness. “There may be cases where you have good intentions of trying to return that person back to that family”, but the family is “why they were homeless in the first place”, said Bob Erlenbusch, a longtime advocate based in Sacramento, California.

There's still this pervasive mythology about family here in the US (and elsewhere). The myth goes that family is better for you than friends, acquaintances or strangers. Obviously, this is not always the case.

The article ends, as is appropriate, by revisiting the Raber story it started with. Basically, he ended up going back to SF and:

Today his circumstances are almost exactly the same as they were before he left. He spends part of the month crashing in a friend’s room in a rundown residential hotel, and the rest bedding down on the sidewalk with a blanket and pillow in the gritty Tenderloin neighborhood, where many other homeless people congregate. He drags his belongings around in a suitcase with broken wheels, and hopes to move into a discarded tent that he found recently.

Discarded, I'm betting, by someone else who got bussed out of San Francisco.

Now, I've noticed that whenever some bleeding-heart suggests actually housing the homeless, the response from some asshole on the other side is always "start with your house." Let me nip that one right in the bud, if any of that type are still reading this: I own two houses. One of them is occupied by a husband and wife who are both disabled and would otherwise have no place to stay. So I do that. I say that not to virtue-signal, which I hate doing, but only to keep that comment from happening here.

I can only help a limited number of people myself. That's why we need to acknowledge that we live in a society, and it's everyone's responsibility.

After all, you may think it won't happen to you. But it could.

7 Like • Comment • View 3 comments

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