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 20 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 22 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.

February 7, 2020 at 12:29am

#975019

Millionaire Blues

Haven't ranted about personal finance issues in a while...

https://www.yahoo.com/lifestyle/mom-became-millionaire-25-170215911.html

How This Mom Became a Millionaire by 25

Becoming millionaires didn't happen overnight, explains Jarman, 27. "We're just working hard and trying to build financial stability for our family. And suddenly we looked back and we realized, 'Oh, look at the money in our bank, all of our assets. Look at what we've accomplished,'" she says.

Look, I'm not going to knock anyone's personal journey, but if you haven't noticed, the last 10 years or so have been rewarding to anyone with any money in the stock market. That doesn't make you a financial genius, even if it did make you rich.

When the Jarmans first graduated from the Marriott School of Business at Brigham Young University in 2015, they had up to $40K in student loans, $10K in assets (like their cars), and about $30K in savings.

So you started out on second base, not third. I mean, I've seen worse.

The Jarmans, who are originally from Southern California, decided to start their careers in Arkansas partially because of the state's low cost of living.

Downside: you're living in Arkansas. Okay, there are worse places. And I agree that's smart: there's one basic, simple, but unforgiving equation for financial success: Income > expenses. Increase income or decrease expenses, and you are on the right track. Do both, and that's a recipe for success.

If you carry any debt, Jarman suggests looking at what is holding the highest interest rate and paying that off first, before you start saving for big expenses, like your kid's college tuition or wedding.

Here's a novel idea: your kid can pay for his or her own college and wedding. Here's an even more novel idea: don't have kids. Last time I checked, the cost of raising one kid to college-age was in the neighborhood of half a million dollars -- and last time I checked was maybe 20 years ago. (That sounds like a lot of money, but it averages out to about $28K a year, which, okay, that's still a lot of money but not out of reach for many people.)

The debt-paying thing, though, I'm totally on board with.

She adds that the biggest mistake she sees is people not living within their means. "My husband and I believe in living below your means," she says. "Just because you’re making this much, it doesn’t mean that you should be spending that much. I think so many people are just working paycheck to paycheck and just spend too much. You need to look very closely at what's a need versus a want."

That's not bad advice. However, it reeks of privilege. Not everyone has the income or opportunities that college-educated west-coasters do. There's a minimum amount that most people have to spend just to survive, and a whole lot of people just don't reach that minimum in income.

"The biggest thing is have a long-term vision," says Jarman. "You can't look at tonight, tomorrow, next week. You have to look five years from now, 10 years from now, 50 years from now, and you just have to constantly remind yourself of the future. Life is just so crazy right now. It’s because we have so many jobs, and I just want to stay home and be with my babies. I'm really close to being able to do that, but it’s been five years of working a ton."

There was another personal-finance article I saw but didn't save, and it tried to combine budgeting with that "mindfulness" nonsense that's all the rage these days. The two philosophies are incompatible. Mindfulness encourages people to live "in the present" (which doesn't actually exist), whereas budgeting uses past spending habits to predict and control future spending habits. I think you can tell which philosophy I adhere to.

I noted recently in here that there's a connection between money gain and weight loss. For the former, it's the inequality I bolded above. For the latter, it's (calories in) < (calories out). Same idea, though the devil's in the details. What articles like this one utterly fail at is understanding that everyone's situation is different. It can give people false hope, even if it gives other people the motivation they need.

There's another article in my queue that speaks to this sort of thing, and I'll get to it eventually. For now, just consider this: being a millionaire isn't that big a deal anymore. I mean, it's better than being a thousandaire, and definitely better than having a negative net worth, but the mere accumulation of $1,000,000, while it seems like a shit-ton of money, isn't enough to buy that house on Easy Street, especially when you're 27 and have a good 50+ years of life expectancy ahead of you. And double especially if you're planning on funding your kids' college and weddings.

Nevertheless, the practices and discipline needed to do that are great life skills to have.

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