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 6 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 8 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 24, 2020 at 12:04am

#991450

Orange You Glad This Wasn't You?

Something a little different today.

Also, Merit Badge Mini-Contest below!

https://medium.com/better-marketing/the-worst-rebrand-in-the-history-of-orange-j...

The Worst Rebrand in the History of Orange Juice
They paid $35 million to then lose $20m in sales

So why am I linking something about orange juice? Because it's a product, and some marketing techniques apply to a broad range of products... like, for instance, your writing.

Not that I know jack squat about any of this. Which is why I read them -- to confirm that not only do I not understand, but I will quite probably never understand. But maybe this will help someone else. Or, hey, maybe you just like orange juice.

In hindsight, it’s easy to see why Tropicana’s 2009 rebrand failed.

No. No, it really isn't. Not to me. And even the author explains, later, that it shouldn't have taken hindsight.

Less than 30 days after launch, they pulled the new design off the shelves and went back to the old one.

Which probably explains why this is the first I'm hearing about this. Coke's "New Coke" fiasco personally affected me. I don't drink orange juice, so I never noticed a rebrand.

The article helpfully includes "before" and "after" pics.

Without even getting into the subjective topics of visual appeal and recognizability, some design flaws practically stare you in the face.

But do they, really?

After the rebrand, the font was thinner, the same color as the remaining text, pushed to the side, and, worst of all, vertical. No matter how great your juice is, if it can’t heal neck pain, don’t force me to tilt my head sideways to read your name.

I have the ability to read text upside down, sideways, backwards, mirror image, upside down mirror image... the latter takes me a bit longer, to be sure. I can also write in all those directions. The sideways text wouldn't even impinge itself upon my consciousness. But hey, not everyone can be a super-genius, I suppose, and you have to market to morons.

What’s more, white font on a yellow background is a weaker contrast than dark green on a white background — especially considering they stuffed it inside the juice glass.

Okay, on that part, I can see their point.

Beautiful design is important, but if it’s not functional, it won’t matter.

Function is more important than form. Well... for me. Engineer, remember? But I can admit that when it comes to trying to sell shit, part of the function is to catch a viewer's eye (or ear in the case of radio). That requires form.

“Historically, we always show the outside of the orange. What was fascinating was that we had never shown the product called the juice.” Really? I mean, it’s juice. Give me a clear symbol of it, and I’m good to go. And what could be clearer than the actual fruit the juice is from?

Somewhere in there, I think, is probably a metaphor for promoting one's writing. I'm not quite sure what it is, but I'm pretty sure it's in there.

The one thing I’ll give them credit for is the cap of the new carton. It had the haptic structure of an actual orange. “We engineered this little squeeze cap so that the notion of squeezing the orange was implied ergonomically.”

Look, I know what "haptic" means, but a few paragraphs ago this author was complaining about jargon (I didn't copy that part, but it's there). Also, making your orange juice cap look like an orange is a surefire way to get me to purchase a competitor's brand. If, that is, I were inclined to buy orange juice.

But then, I freely admit I'm not a typical consumer.

So, anyway, like I said, maybe someone else can make more sense out of this. And hell, why don't we make it a mini-contest?

Merit Badge Mini-Contest!

Know any good (by which I mean bad) marketing failures? Comment below. The one I like best will earn its author a Merit Badge. Two restrictions today: 1) Don't talk about New Coke. I know about New Coke. I suffered through the Great Coke Crisis of 1985, and I have no wish to relive those horrid months. 2) Don't use this blog as an example of a marketing failure (I'm including this because I'm the sort of asshole who would do something like that, and besides, I'm not trying to sell anything here. Quite the opposite.)

If you can't think of one, it's not cheating to use Google. Mostly I just want to understand this marketing thing better, and, just like they kept showing us the newsreel footage of the Tacoma Narrows Bridge collapse

as a warning in engineering school, I think we can learn from failures as much as from successes. It's best if it's not our own failure.

6 Like • Comment • View 9 comments

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