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 2 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 4 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.

August 6, 2022 at 10:44am

#1036176

Can't Budge It

Posting late today because The Sandman dropped yesterday, so of course I had to start watching it. I say "start" because I haven't finished the series yet. I'd say "don't you dare spoil it if you have" but hell, I've only read the graphic novel sixteen times.

But today's article isn't about that. It's about money, which I hope Gaiman makes a lot of from this (and that's the worst segué in the history of segués)

Budget Culture and the Dave Ramseyfication of Money
How financial "experts" ruined our relationship with money

First of all, Ramsey can bite me. Sure, he has some good ideas, but they come from a distasteful framework.

Unlike previous editions, which have focused on reader Q&A, this edition, from personal finance writer Dana Miranda, focuses on a particularly noxious ideology about money: where it comes from, how we internalize it, and how it haunts us for the rest of our adult lives.

I haven't read these "previous editions," so I probably lack some context here.

Before Ramsey’s call-in talk show (“The Dave Ramsey Show”) and his book (Financial Peace) financial advice was largely for people who were already rich. It went straight to stock picks and skipped what to do if you had credit card debt or lived precariously from paycheck to paycheck — assuming people were in those situations because they didn’t care about financial education.

It is true that before you even think about investments, you need to get your spending and debt under control. This was easier in the past.

Other budding financial experts saw the need for similar advice that dropped Ramsey’s religious exclusivity, and a new “everyman” niche in personal finance emerged around the turn of the century. It ballooned in the wake of the 2008 financial crisis and the popularity of personal blogs, where so-called everyday millionaires could chronicle their journeys out of debt and into the middle class.

To be clear, I benefited from some of that advice, myself. The trick is to adapt whatever they're saying to your own situation—if you can. No single process works for everyone. It's important to do something, though. You know those stories about lottery winners and heirs who blow through their windfalls and end up in worse situations? That's because if you don't know squattly-dick about how to handle money in the first place, no amount of money will fix your life. I have another article about that sort of thing in my queue, so eventually I'll get around to expanding on that topic.

Ramsey and his successors put the “personal” into finance so anyone could find success through a new set of rules made just for us, like paying down debt, starting a side hustle and — above all — making a budget.

There's nothing inherently wrong or scary about budgeting, but this article is about to tell us otherwise.

This promise appealed directly to the work ethic of middle America: You can get rich with steady work and self control. The marriage of personal finance and self improvement — the Rich Dad Poor Dad, Millionaire Next Door, Finish Rich ethos — set a tone for our current dominant paradigm, which I’ve come to call budget culture.

If hard work were all it took to become financially secure, sharecroppers would be the richest people on the planet.

Budget culture is the damaging set of beliefs around money that rewards restriction and deprivation — much like diet culture does for food and bodies — and promotes an unhealthy and fantastical ideal of financial success.

I've noted before the parallels between diets and budgets.

The simplest problem with this approach is that budgeting, like dieting, doesn’t work. Restriction and deprivation are unsustainable; accurately tracking spending is hard; income fluctuates, and costs don’t operate on a perfect monthly reset. Budgets don’t account for the way lives work, so they’re hard to keep in our lives consistently.

It doesn't work for everyone. Author may be projecting.

The broader problem with budget culture is its emphasis on individual responsibility and insistence on ignoring the varying levels of access and privilege in our world. It vilifies and oppresses anyone who doesn’t live up to the ideal, regardless of their circumstances. And that ideal is, unsurprisingly, rooted in maleness and whiteness in the way many of our cultural ideals are.

It is true that individual responsibility is often over-emphasized. There are myriad factors outside our control that need to be accounted for (pun intended). Those factors can only be worse if you're already in a marginalized group.

The only sure way to make money is to have money, but no personal finance expert wants to admit their wealth is built on anything other than a solid foundation of hard work and self-control — not their degrees in finance, Ivy League educations, middle class upbringings… or their ability to sell you a fantasy.

It is also true that any self-help book—not just the personal finance ones—are mostly there to self-help the author get rich. Me? I'm not making any money from this. That in itself is a privilege, because I don't have to, but neither am I obligated to censor myself or present an approved point of view.

Anyway, the article has some other interesting points, but I'll stop here. Back to The Sandman...

4 Like • Comment • View 3 comments

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