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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 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 10 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.

January 17, 2024 at 9:45am

#1062540

In Decision

Here's a pretty recent one from Cracked. Mostly just because I felt like sharing it.

4 Times Corporations Invested in the Exact Wrong Thing

The title is maybe a bit misleading, but close enough.

Making decisions has to be one of my absolute least favorite things to do. That’s why I try to avoid it at all costs.

Would it help to know that your decision is already made, but you just haven't caught up to that moment in time yet?

And the more decisions matter, the more they suck. Picking a cereal to start your day with is a decision, but given the general futility of life, who really cares? Realizing halfway through a bowl of Cap’n Crunch that you really would have preferred Froot Loops is just another thin cut delivered by the blade of life.

I avoid this by not having cereal in the house.

It’s when you’re faced with decisions that really influence the future of your life or business that you get to experience the full cocktail of stress and anxiety.

Hence the relevance to the article's content. Though sometimes the wrong decision turns out to be not deciding at all.

4. Sony Backing Betamax

Given that plenty of people under 30 probably don’t even remember VHS tapes, I’m going to wager that Betamax rings even less bells. It’s an old, cursed technology, that barely made the cut for my 33-year-old brain, much less Zoomers’.

We really need to teach history better.

Anyway, yeah, so Sony backed the losing technology. They're still around, though, unlike the next fool:

3. Borders Gives Amazon Its Business

The biggest mistakes they made? First was not investing in or offering e-books, and the second was outsourcing all their online book sales to a little company called Amazon.

Yeah, I'm gonna call that one of the worst mistakes of all time.

2. Kodak Doesn’t Go Digital

This one's pretty well-known.

In 1975, a Kodak engineer named Steve Sasson invented an early digital camera that could record images electronically onto tape and display them on a TV screen. He showed it to executives, who gave it a resounding thumbs down, because it would ruin their ability to sell film, a big part of their business.

So someone else went to market with it, and the tech steadily improved, ruining their ability to sell film. If only they could have seen the writing on the wall. (Okay, that might be an obscure reference.)

1. JCPenney Respects Its Customers’ Intelligence

I'd have titled this "JCPenney Overestimates Its Customers' Intelligence." I mentioned the 1/3-pound burger fiasco a few days ago in here, and that people were so abysmally ignorant that they thought it was a worse deal than a quarter-pounder because, clearly, 3 is less than 4. Well, this is similar.

It turns out that they do not want a $40 waffle maker, they want a $40 waffle maker they can tell people was worth $80, and if they don’t get it, they will figuratively burn your store to the ground. Sales dropped 20 percent, and one ex-customer with a real peach of a brain explained, “If I don’t get a special discount, it’s not worth the trip.”

No, I'm not calling YOU stupid. Anyone who reads this blog is automatically smart, attractive, and talented, if only by association. But goddamn, the vast amorphous bulk of consumers are idiots to the core.

Anyway, I'd add one more to this list: Sears. They built a sales empire on mail-order, sending catalogs to the farthest reaches of the US and Canada, and probably beyond, I don't know. At one point, they were mail-ordering houses. I'm not kidding. They got so huge that they built and occupied what was then the tallest building in the world.

And then the internet came along, and they didn't keep up.

There are probably lots more, of course. Most decisions are going to turn out to be bad ones, or at least neutral, which is why people like me and this article's author avoid making them whenever possible. Which is also going to be a bad decision, but whatever, at least then it's not my fault.

8 Like • Comment • View 1 comment

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