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 15 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 16 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.

April 16, 2020 at 12:30am

#981227

Again with the Millions

One meeeeelion dollars.

*starts to touch pinky to lips, remembers coronavirus, shrugs*

https://www.thecut.com/2020/03/im-a-secret-millionaire.html

‘I Inherited Millions and I’m Hiding It From Everyone’

So far, smart. People find out you have money, they'll try to find ways to part you from it.

But for the wealthiest 5 percent of American households, the average inheritance is $1.1 million, according to a 2014 report from the Federal Reserve.

I can't be arsed to click on that link, but "the wealthiest 5 percent" covers a really broad range of net worth. I'd be interested in how they figured that average, except it's too much work. Whatever. Let's round that down to One Meeeelion Dollars for the sake of argument.

But unlike most heirs, she hasn’t touched the money at all — or told anyone about it.

You know how you're always reading about heirs and/or lottery winners who blow through the windfall and end up broker than when they started? That's because smart people a) don't blow through the windfall b) don't tell too many people about it and c) aren't usually interesting enough to have articles written about them. Fortunately, this one's a (c) exception. Well, she's still not very interesting, but she had an article written about her, so there's that.

I was raised to not flaunt money. My parents always told me it was a stupid thing to do, and we lived a very middle-class lifestyle.

"Middle-class" covers a wide range, too. If you're talking lifestyle, it could be anything from a modest three-bedroom house and a Ford Fuckus to a 7 bedroom, 5 bath McMansion with 3 SUVs in the garage.

The problem is, most people think that the latter indicates greater wealth than the former. It does not. It often indicates higher debt.

They were both self-employed — my dad worked in alternative medicine, and my mom ran a daycare business.

"...and our Housefinders budget is $3.2 million."

My parents left me a seven-figure inheritance, in cash.

Another potentially misleading statement. There is a huge, enormous difference between whether those seven figures start with 1000 and if they start with 9000. The former will yield a safe withdrawal rate of about $40K a year (how much you can pull out, while allowing the principal to continue to grow to keep up with inflation). And that's nice; a lot of people live on that. The latter is closer to $360K a year, which would fund some truly epic vacations.

But, okay, elsewhere it's implied that it was closer to One Meeelion Dollars, so again, I'm going to stick with that.

I made all of my investment decisions on my own, without an investment adviser, from research I’d done while managing my own personal portfolio.

That's great if you have the aptitude and desire for it. I'd venture that most people don't. While there's no shame in hiring a financial adviser, be aware that their fees cut into gains. But if the alternative is "no gains," then yes, you come out ahead.

By the time I inherited money, I already had six figures in my portfolio that I’d accumulated independently.

And for the third time: this is misleading. 100,000 is an incredibly different number than 900,000.

But fine; that's how people talk sideways about money, instead of coming right out and saying "I had a portfolio worth 250K," they'll just say "six figures," even when they're being anonymous like this person is. It's really remarkable. People will talk about their hemorrhoids, but not their money.

But I don't talk about mine, either, so I suppose I can't be too surprised.

I’ve now had this money for three years. It was a lot to handle at first, and there was a steep learning curve, but I’ve already grown the portfolio by 50 percent.

Obviously, this was written (if not dated) before the current pandemic and its associated recession. Fifty percent in three years is pretty good at any time, but pick any three years of the teenies' bull market and you could probably achieve that by picking stocks at random. As a general rule of thumb, money fully invested in the stock market can double every nine years or so - of course, that depends on whether there's a recession in those seven years or not, but on average, that's been the case.

Everyone has different things — what I lost, many other people still have.

And this is important. Money is good, but some people seem to value family more.

My parents had a non-flashy, stealth-wealth lifestyle, and I’m the same way. I don’t even have a car. I don’t wear any labels. I ride a bike everywhere. I make my lunches. Nothing I do would elicit people to think, “How does she afford this?”

I'm of the opinion that those who do flaunt wealth generally don't actually have it; they just want to promote the appearance of having it. Like I said above. I guess it's all about what's important to you, and for me, money is more important than perceived social status.

That doesn't mean I'm not going to buy quality stuff (it's cheaper in the long run), but the only "label" I wear is Levi's.

I think the biggest thing I’ve learned from this experience, and working for over ten years now, is that your job doesn’t have as much of a relationship to your net worth as many people think.

This is really important, I think. It doesn't matter if you make "six figures" in a year if you spend "six figures" plus x, where x is any number, as per algebra. You end up with more stuff, maybe, but come the coronapocalypse, you found yourself just as broke as someone who was scraping by on rideshare tips.

But I see it conflated in articles all the time. A "millionaire" is someone with at least a One Meeelion Dollar net worth, not someone who takes home that amount in a year. The latter person might well be spending most of it on hookers and blow, and wasting the rest, instead of building net worth.

And why is net worth important? Because as we've seen over the last couple of months, your job - your entire industry - could collapse at any moment, and it's clear that significant government handouts are reserved for the likes of airlines and banks, not their customers. Meanwhile, we were encouraged to spend every cent we made, and then some (by going into debt), to fuel the economy. But people with some money saved up are weathering this crisis much better than the rest of us.

And if they're smart, they're not bragging about it.

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