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 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.

December 26, 2023 at 9:07am

#1061420

Biggerer

Another Cracked article, once again illustrating the randomness of my random number generator; I have far more links from other sources.

5 Things Getting Bigger When No One’s Looking

I am resisting the obvious joke. Resisting with all my might. Resisting hard, you might say. Can this article also resist?

You turn around and see something behind you. Later, you turn around again, and it’s bigger now. This means either it’s following you, or it has physically grown in size.

Or both. Why not both?

5. Spiders

You'll want to click on the link for the illustration of what a spider might look like.

Bigger spiders help cities, since bigger spiders can gobble more bugs. They also serve as more food for birds. Of course, any change in the food web can spark unpredictable consequences, and spiders eating too many bugs or feeding too many birds could be disastrous, but a thorough analysis of the side effects leads the scientists to conclude that, yes, giant spiders are good news, everyone.

Spider-like typing detected.

4. Eyes

Including spider eyes?

One part of your body grows for your entire life. We’re not talking about fat — many parts of your body grow when you eat more, but one part grows regardless.

Almost did the obvious. Almost.

We have one single confirmed example of an organ that grows, forever. It’s the lens of your eye. Each lens grows by 1.38 milligrams a year, and while that sounds like nothing, we have to take into account just how small lenses are to begin with. Between early adulthood and when you die, you can expect each lens to double in size.

Unless, of course, you get off the growth treadmill by having cataract surgery. Which reminds me, it's been two years now.

3. Finland

As sea levels rise, countries all over the world have to reckon with losing some land. This loss will range from an inconvenience to apocalyptic.

Nah, they'll just redefine the border as being 100 yards out from shore.

Finland is an exception. Finland gains 7 square kilometers (3 square miles) a year. No, it’s not because they’re so militant about combatting climate change that they’ve somehow convinced the sea to recede. They’re instead experiencing a phenomenon called post-glacial rebound.

The opposite of shrinkage?

2. Breasts

...

...

...

I'm sorry, did you say something?

1. Zoo Animals

Maybe the animals are under the influence of endocrine-disrupting chemicals, or maybe they’ve been infected with some obesity-causing virus. Maybe that’s also why we’re getting fatter, rather than the traditional explanation of overeating.

But that explanation offends puritans, who are certain that everything from obesity to climate change is our fault because we've sinned.

And there it is... the whole article and my commentary, without the obvious joke. I feel shafted.

6 Like • Comment • View 6 comments

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