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 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 8, 2022 at 12:01am

#1034851

Negative Is Positive

You want me to "be positive?" Okay... I'm positive we're boned.

A psychologist explains why negativity dominates your daily thoughts, and what to do about it
Brain science offers strategies that go beyond positive affirmations to create more lasting change.

But I thought we were supposed to accept ourselves as we were? Or is that bullshit, too?

We’ve all been there, mired in the throes of pessimism when life just doesn’t seem to be going our way.

Hey, be fair, now. Life usually seems to go my way (with notable exceptions, as with everyone) and I still cling to pessimism like a koala to a eucalyptus tree.

For the lucky ones, this entirely natural, though disconcerting, feeling ebbs and flows, ultimately dissolving into a more balanced, healthier state of mind. For hapless others, though, the extreme negative thoughts and ideation can overwhelm—even becoming “who they are.”

Damn right it's who I am.

At worst, it can be deadly, as one peer-reviewed study found that “people who are strongly pessimistic about the future are at greater risk of dying earlier than those who are not pessimists.”

And this is bad... how?

“Even though, at times, we may feel as though we are not, we are the ones in control of our thoughts. We can choose not only what we think about but how we think about it.”

Yeah, no, that makes no sense at all.

Vermani says that we need to learn to actually challenge our cognitive distortions and negative thoughts, and question how real or accurate they are.

I'm not going for accuracy, here. I'm going for pleasure. If I expect the worst, then one of two things will happen:

1) The worst will happen, and I get the immense pleasure of Being Right;

2) Something better happens; it is, by definition, better than what I expected, and that pleases me as well.

I wonder if these researchers took into account that, for some of us, pessimism gives us a deeper, more realistic happiness?

According to Digital.com marketing and small business expert Dennis Consorte, it’s inevitable that we’re thinking negatively because, as a society, we’re surrounded by it. “The media sensationalizes stories to drive clicks and views,” he says with lament.

Oh, it's far worse than that (you expected me to say that, didn't you).

Advertising (like Mr. Marketing up there creates) takes advantage of this psychology, too. In its purest form, it manufactures a void in our life that can only be filled by Shiny Thing. After we obtain Shiny Thing, it loses its shine and so we move on to the next Shiny Thing.

This is one reason I avoid ads like my cat avoids water.

According to transformational guide Jaime Haas, it is downright critical to identify harmful circumstances that were imprinted before the rational mind was even formed in early adolescence and to extrapolate and understand how those situations likely formed patterns that could be limiting one’s present-day life.

Give money to psychologists, and you'll be happier! It's a Shiny Thing!

Hass says Negative Manifestation Compulsion (NMC) is controllable. She says that NMC often occurs as a sequence of negative thoughts in close succession, compounding upon each other and ultimately leading you away from, rather than toward, your goals.

My solution? Don't have goals.

“Many self-help ‘gurus’ offer advice that is one-sided,” Haas says. “They assert that positive affirmations are key to solve our problems, but I’m here to tell you there’s nothing that’s further from the truth. Optimism needs an authentic foundation in order to be upheld, realized, and certainly sustained..."

Well. On that, we can agree.

Vermani offers a few other relatively easy ways to help gain control of negative thoughts. First and foremost, she suggests surrounding yourself with people who enhance, support, and inspire you, lifting you up and helping you see the good among the challenges in life.

See? That. Right there. "Surrounding myself with people" of any sort will quickly spiral me into despair (which, to be clear, I rarely experience). This may be fine advice for extroverts, but it absolutely would not work for me. "Waltz, you shouldn't make such negative statements!" "Okay, I'm positive it wouldn't work for me."

Don't get me wrong - I have friends, and I appreciate them. There are some people I can be around longer than others. But eventually I need to retreat. It's also not, as some might believe, because I feel like I'm "too good" for them. Quite the contrary. I start to sense that my company is starting to drag (even when I'm not being pessimistic), and I don't want to inflict myself on others any longer than is necessary.

This includes minimizing sources of stress by setting healthy boundaries around exposure to critical, combative, and negative people—as well as the endless news cycles. She also recommends actively and concertedly letting go of negative self-perceptions, limiting beliefs, thoughts of being not good enough, and conditions you place on yourself in order to be happy.

I mean, I do that first bit, kind of. I avoid social media. Most of the articles I seek out involves science (like this one), though I can't help but be exposed to news of wars, famines, pestilences, and deaths. But when it comes to thinking I'm not good enough, I assure you, it is indeed the case that I am not. And I'm okay with that. Other people might not be okay with me not being good enough, or maybe they believe I only think I'm not, but that's not my problem.

And as I've said before, happiness isn't a goal; it's a result of other things. I achieve it by knowing my limitations and only setting easy targets for myself. Well, okay, I do have the completely impossible plan to visit every brewery in the US, but that's partly so I don't have to think about trying to do anything else. And mostly so I have an excuse to drink beer.

In conclusion, maybe the stuff in this article works for the majority of people. I wouldn't know. But I (mostly) accept myself for who I am, and I see no point in changing.

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