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').

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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.

April 22, 2021 at 12:02am

#1008945

I Don't Buy It

There's a whole lot wrong with the phrase, "Money can't buy happiness," starting with the assumption that happiness is some sort of worthwhile goal. So here's an article about it that's also wrong.

How to Buy Happiness
The joys of money are nothing without other people.

To quote the Beatles, "Money can't buy me love." To instead quote the Beatles, "Money don't get everything, it's true, but what it don't get, I can't use."

In 2010, two Nobel laureates in economics published a paper that created a tidal wave of interest both inside and outside academia. With careful data analysis, the researchers showed that people believe the quality of their lives will increase as they earn more, and their feelings do improve with additional money at low income levels. But the well-being they experience flattens out at around $75,000 in annual income (about $92,000 in today’s dollars).

Nobel laureates: appeal to authority. A paper: oh, boy, a paper. Data analysis: wow, then it must be true.

I can't be arsed to look at the actual paper (which I've been hearing about for 11 years now), but when someone's making more than $100K, they usually have other stressors that might limit their sense of well-being. In other words, once again, correlation isn't causation.

This January, another economist published a new paper on the subject that found that even beyond that income level, well-being continues to rise.

Dueling papers!

The lesson remains the same as it was a decade ago: At low levels, money improves well-being. Once you earn a solid living, however, a billionaire is not likely to be any happier than you are.

Yeah, but again, why is happiness the goal? What if I don't want to be happy, but just want to fly around in a Gulfstream and sail around in a private yacht?

Yet for the most part, this truth remains hard for people to grasp. Americans work and earn and act as if becoming richer will automatically raise our happiness, no matter how rich we might get. When it comes to money and happiness, there is a glitch in our psychological code.

I'm not sure that's the case. We're talking about Americans here. No matter how much money you make per year, no matter how much savings you manage to accumulate, one medical event can wipe it all out entirely, leaving you in poverty and/or debt. Money is a cushion against such things; the more money you have, the more likely you are to financially survive one trip to the hospital.

In other words, money is security. It's not really about happiness.

Research shows that how the wealthier among us spend their money makes all the difference for their well-being. Specifically, spending money to have experiences, buying time, and giving money away to help others all reliably raise happiness. Thus, if you have a little excess income, it’s best to use it on those three things.

[citation needed]

The key factor connecting all those approaches is other people. If you buy an experience, whether it be a vacation or just a dinner out, you can raise your happiness if you share it with someone you love.

Eh, no, not really. Adding other people into the mix means you have to recognize their desires. Say you go to Vegas with a friend. You want to gamble; they want to see a comedy show. So either one of you does something you don't want to do, or you each go your own way, in which case why bother with the friend? I mean, sure, traveling with friends can be fun, but for the most part, I travel alone so I don't have to have endless discussions over what to do next, what to eat, when to sleep, etc.

Friends and family are two key ingredients in well-being, and fun experiences with these people give us sweet memories we can enjoy for the rest of our lives—unlike the designer shoes that will wear out or go out of style.

Designer shoes may go out of style, but friends end up ghosting you, and as for family, I realized recently that my definition of "family" is "the people who, no matter what you do, it's never good enough for them and they'll be disappointed in you." Nah, I'd rather have money.

Likewise, if you pay someone to do something time-consuming that you don’t like to do (for example, cutting your yard), and don’t waste the time you gain on unpleasant things like doom-scrolling on social media, you can get a happiness boost by spending those extra hours with others.

I play video games (the single-player kind) while someone that I'm paying is mowing my lawn. That makes me feel good.

Left to our urges and natural desires, we can get stuck in a cycle of dissatisfaction, in which we work, earn, buy, and hope to finally get happier.

Yeah, well, at least part of the problem is a constant barrage of advertisements designed to make us dissatisfied with our lives, and if we only bought This Thing, we'd surely be happier. I know I've said this before, but that makes people think that happiness is having what you want; I say that happiness (if it's even a worthwhile goal in the first place) is wanting what you already have.

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