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 10 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 12 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.

February 17, 2024 at 9:04am

#1064327

Inside Voice

Bit out of the ordinary for me today—a link sent by a friend:

Some People of Color Don't Like the Outdoors and That's Okay

As I, too, shun the not-so-great outdoors, I found great wisdom in these words.

The outdoor industry has spent hundreds of millions in marketing dollars to convince you that everyone belongs outside.

To be fair, lots of industries spend lots of money trying to convince us that everyone needs their products.

That simply isn’t true. Some People of Color aren’t outside for very valid reasons.

As far as I'm concerned, every reason is a valid reason, but I can only imagine how much worse it must be when you have to be concerned that racist rednecks love to be outdoors. With guns.

Some don’t like the outdoors and that’s okay.

"I don't like it" is a valid reason. So is "I don't want to."

They don’t need you to convert them. Unless, maybe, you’ve found the perfect beginner-friendly hike that ends in a waterfall and free universal healthcare.

Okay, that bit made me laugh.

But failing that, here are eight reasons why you should leave your friends and family alone.

Oddly, like Cracked (which this isn't from), it's a countdown list. I won't be covering everything.

Some of us occupy entirely different realities from our parents. For us, sleeping on the ground is an adventure. For them, sleeping on the ground is elective poverty and strange (or a bad memory). Why would they pay to do that?

I've gone on about this sort of thing before. You live in a nice house, and you want to experience two weeks of homelessness? With bears and snakes? Okay, you do you, but count me out.

Some of your friends aren’t interested in hiking or climbing because they can’t afford to get hurt and miss work. They also can’t afford medical care.

Laaaaand... of the... freeeeeee!

And yes, you can get hurt at home, too. But I'm pretty sure the chances of that go up in areas of uncertain footing, wild animals, and stinging insects.

4. They don’t want to get their hair wet

This section turned out to be important for me to read, because I was mostly unaware of this aspect of culture.

3. They don’t have the money

We like to pretend that being outside doesn’t cost a thing. And it’s weird. Stop that.

While poverty looks different for different people, this point is something to remember when considering a trip to the not-inside. Proper gear (which I'm sure would help reduce the chance of injury, as above) can get expensive. Even those not experiencing poverty might have better (to them) things to spend money on.

Why do we pretend that the outdoors costs nothing—that you are required to bring nothing but yourself? Then in the next breath we shame hikers who don’t bring reusable water bottles, moisture wicking technical clothing, wool base layers, wind and water resistant shells, puffers, GPS, a trail map subscription with downloaded maps, a paper map and compass, etc.

I read an article about someone who died on Mount Washington in New Hampshire last month. The article described him as an "experienced hiker." Now, I certainly don't claim to be an expert on mountaineering, though, yes, I've hiked mountains, and experienced firsthand how quickly the weather can turn on you there. And I've read that Mount Washington is the worst of the bunch, being well-known for having logged the highest recorded non-cyclonic wind speed of any point on Earth. So my first thought was "Funny; I thought 'experienced hikers' don't mess with Mount Washington in fucking January."

It’s winter now and social media is full of photos and videos of happy people skiing, snowshoeing, tubing and snowboarding.

One of those things is not like the others...

Anyway, while a lot of the cultural reasoning doesn't apply to me, and to be fair, I was privileged enough to spend a lot of time outdoors as a child and young adult before deciding I just didn't like it, I appreciate the article's perspective. And to be clear, it's not "being outside" that I dislike; it's being farther than a short walk from civilization.

Now watch, some marketing guru is going to try to take all the stuff in that article into account the next time they need to market a line of rugged-yet-fashionable outdoor wear that'll be lucky to last for a year.

3 Like • Comment • View 5 comments

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