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!

a day 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

a day 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 10, 2019 at 12:23am

#963997

What A Fool Believes

I don't know why I keep going back to this guy's blog.

https://dariusforoux.com/if-you-can-believe-you-can-achieve/

If You Can Believe It, You Can Achieve It

It's generally the epitome of corporate-style motivational affirmations. I'm linking this particular one because he talks about writing, and, well, here we are on a writing site.

I think the world has no shortage of motivational articles, books, videos, or Facebook posts.

You know what motivational quotes make me want to do? Give up. That's what.

Belief is a practical instrument that you can use to shape reality.

No. No, it really isn't. You can use it to shape your attitudes about reality, but reality is reality, by definition, q.e.d., etc.

Have you ever considered that you decide what you believe? Not your friends, colleagues, family, or even the media.

You'd be surprised. There's a whole industry devoted to making sure we believe what someone wants us to believe. Often these beliefs are contradictory. The amazing thing about being human is that we can believe contradictory things simultaneously. This is both a benefit and a detriment.

It comes down to one thing: Do you believe that you can, or can’t live the life you want?

I am living the life that I want. The thing is, it's not necessarily the life that I wanted 10, 20, 30, etc. years ago. Over time, I've adjusted my expectations. I find that that, more than belief, more than motivation, more than anything else, is the secret to happiness. Adjusting expectations meant I could stop, for example, believing that I could invent a flying car, or even own one.

So now we come to the writing part:

I always wanted to become a writer. Back in school, I wrote poems for my girlfriend. It was kind of pathetic, but hey, she enjoyed my weird poems. But all jokes aside, I loved to read and enjoyed putting my thoughts on paper.

But no, every adult in my life was trying to scare me to death. “You can’t make a living as a writer.” They were probably right. It’s not easy to make a living by writing.

True enough. My parents wisely steered me away from creative pursuits so I could make a decent living. "Wisely" for two reasons: 1) it's hard and 2) I'm not very creative.

So what? Everything that’s worth it is hard. I didn’t realize that back then.

That's simply not true, in my experience. The things I appreciate most are the things I didn't work hard for. The harder I work for something, the less satisfying it is - because by the time I'm done putting in the work, it's never as bright as my expectations were.

But, okay, I'll accept that some people feel that way. Fine.

Granted, I’m not fully paying the bills by writing yet. I run a business and do consulting. But I do make some money with my blog. And that’s pretty good to know for the seventeen-year-old version of me.

Meanwhile, I've never made money from writing. Well, that's not entirely true; I suppose Gift Points count, sort of, as do Amazon gift cards - because you can buy stuff with them. I'm not sure I want to make money from blogging; I'd have to take the time to "build a brand" or whatever, and I'd be limited in what I could say.

Sure would be nice to get published, though, but I'm still not convinced that if I just believe hard enough, it'll happen.

Because my mind can conceive it. And if your mind can conceive it, you can achieve it.

That’s not a motivational quote. That’s a fact.

Do you believe it?

No.

My mind can conceive that I'll have a romantic encounter with Halle Berry. It may or may not be that I've put a lot of mental energy into this belief, but let's say for the sake of argument that I have. Will it happen?

No.

My mind can conceive of faster-than-light travel. Could be we'll discover that, one day, but it won't be me doing the discovering, and unless it happens soon, I won't even be around to enjoy the results.

My mind can conceive all sorts of fantastic scenarios, but the closest I'll ever be is if I write a story about them. Worlds made of gold. Time travel. Honest politicians. Flying pigs. Talking cats.

Look, I can accept that attitude makes a difference. But some things are simply unattainable, on a practical level. I've done many things I never thought I'd do, like horseback riding through a jungle and viewing an active volcano. I've also not done many, many more things that I thought I would do, because no, it's not entirely up to me. Sometimes things happen, or they don't happen; you can't make someone fall in love with you, for example.

So I'm not sure why I waste my time with this kind of mind-massage. Maybe I keep hoping that I'll finally read something that is actually applicable to me. Maybe I'm just fascinated by how people - people who are not me, that is - can seem to put whatever New Age word salad on a screen or a page, and get read, while I languish in obscurity.

Obscurity has its benefits, of course, so I can embrace it. At least no one picks apart my verbal vomit, quote by quote, on their own blogs.

Until, at least, someone's analytics finally point them to this blog, at which point I'm probably going to be in big trouble.

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