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:

Turkey DrumStik

about 9 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

October 14, 2018 at 12:48am

#943386

The Wisdom of the Ages

Time to get into Grumpy Old Man Mode. I'm going to go get my cane so I can shake it at appropriate points.

https://www.becomingminimalist.com/craft/

This article is unmitigated claptrap.

Here are nine ways to begin crafting a life you don’t need to escape:

1. Make Relationships a Priority.

Wrong!

*shakes cane*

Make money a priority.

People will let you down. All of them, eventually. Not all the time, and it's certainly possible to have friends and family, but know that they will let you down. This is like living in Florida - perfectly fine most of the time, if a bit humid and surrounded by crazies, but eventually you'll catch a hurricane. It's perfectly natural, and you'll recover, but it sucks when it happens.

Money, handled properly and intelligently, will never let you down.

2. Remove Unneeded Possessions from Your Home and Life.

Spoken like someone who's never had to worry about replacing shit.

My parents lived through the Great Depression. They never, ever discarded anything that might be useful. My dad's workshop in the basement contained an assortment of screws, nails, springs, hinges, etc. to put hardware stores to shame. When something broke, he'd use the scraps to fix it.

Once you throw away something, and it turns out you need it again, you have to spend money to get it. This is not handling money intelligently. I'm not saying to keep around old pizza boxes or every pasta sauce bottle, but if I had to buy a new tube of superglue every time I needed it because I threw it away after using it once, I'd go broke.

There's also the life cycle thing - I have a working hypothesis that everything goes through three stages: First, it's brand-new and in high demand. Then, for a while (the timing depending on the thing in question), it's commonplace, ordinary, and for most people, disposable. Finally, it's valuable again because it's old and most people have thrown it away. Keep something long enough, and you bridge right over the second stage. It doesn't matter whether it's a home, a car, a soda bottle, a comic book, a stamp, or some random item picked up at a store - at some point, it'll be valuable again.

And finally, I have a crap memory, so "stuff" is the only way I can remember that something happened and how it made me feel at the time.

3. Make Your Work Your Job.

"When your work is also your job, you have achieved a sweet spot in life."

Yeah, and when you're riding a unicorn and smelling its awesome glitter rainbow farts, you've also achieved a sweet spot in life. Utter nonsense. The only purpose of a job is to make money. Save as much as you possibly can, and you'll end up in a place where you no longer need a job. That is the sweet spot in life.

4. Or, See Your Job as Part of Your Work.

Okay, I'll relent a little bit here. A lot of jobs are psychologically damaging. Find one you don't hate, if you can.

5. Guard Your Time.

Yeah, this directly contradicts their point #1. Relationships mean you're running around dealing with other peoples' problems. A little bit of that is okay. Too much, and you have no time to deal with you own crap.

6. Take Care of Yourself.

"What matters at the end of our life is not the house we lived in, the car we drove, or the possessions we purchased. What will matter in the end is how we treated others."

I'll give them this one. Intelligent use of money doesn't mean spending it on things; further, accumulating possessions thinking they'll make you happy never works. And it is important to treat other people well. Still, it doesn't work to deny ourselves the occasional splurge. And I can make the argument that what matters at the end of your life is that you lived it on your terms, not your mother's, and certainly not those of some random new-agey writer on the internet (or some random curmudgeon like me).

7. Appreciate Your Season in Life.

I'll "season" you, you little whippersnapper.

*shakes cane*

8. Understand the Reality of Trials in Life.

Avoid drama at all costs. Remove from your life every possible source of unnecessary drama, even if that source is family. Especially if that source is family. You'll still have "trials," but they'll be a lot easier to deal with if you're not also dealing with your sister's ex-husband's shenanigans or bailing your brother out of jail again.

9. Find Happiness in Your Every Day.

Crap. The best way to be happy is to not think about happiness or try to be happy. Then it comes easily. Forget "mindfulness," forget about trying to "live in the present" (there is no such thing as the present, and the future is unknowable; consequently, only the past is real). Take joy in the simple things, like watching kittens play or seeing other people suffer karmic justice.

Also, I've found that everybody who talks about needing to "escape" has kids. I don't have kids, so I don't feel the need to escape. I like to travel, sure, but I go to places, not escape from the place where I'm living. You can talk about all the joy you get from your kids all you want, but I'm not buying it - I see the problems they cause.

In conclusion, don't be a dick, but also don't be a pushover. Make your own choices. And stay off my lawn.

*shakes cane*

2 Like • Comment • View 3 comments

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