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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10 Most Recent Blog Comments:

Write_Mikey_Write!

a day 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

a day 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').

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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.

June 20, 2019 at 12:45am

#961207

Baggage

You are going on a long trip. I know you’ll take toiletries and clothes. What else has to go with you? Give me a list of 3 things you must take with you and explain why these are necessary.

Ah... welcome to my life.

Well, not my entire life, obviously. I only go on long trips once or twice a year. I know, I know... "only."

So let me describe some of my necessities for such trips.

1. CPAP machine

This should be self-explanatory, but, short version, so I don't die.

Long version: Like many other people, I suffer from sleep apnea. It was probably a contributing cause for the heart attack I had 5 years ago, along with snow-shoveling and other hard work. (I'm certain that pizza, cigars, booze, pretzels, and a sedentary lifestyle had absolutely nothing to do with it.) So after I recovered - which didn't take long, thanks to modern medicine that I can no longer afford - I did the whole sleep study thing and ended up with a CPAP machine.

The improvement happened overnight. Literally. Only real downside is that I can never sleep anywhere that doesn't have an electrical outlet. This is not a problem for me, as I utterly despise camping, and this gives me the perfect excuse to never suffer through another camping trip. Another, more minor, downside is that it relies on a humidifier, and the humidifier requires distilled water. I can bring some with me, but for any extended trip I need to know there's a store somewhere that sells the stuff.

Incidentally, I blew a lot of money on a home generator for the specific reason that if there's no power, I can't use the CPAP, and the power has been known to go out here for a week at a time - and I live in a city. Okay, it was also to keep the foundation drain sump pump running and maybe also so the internet stays up, but if anyone asks, it's for medical reasons.

I don't know what I'm going to do in Scotland - the outlets are different there, and I can't assume I can just pick up distilled water at the corner store. Nor can I schlep three gallons of the stuff in my luggage. Well, I'll figure something out.

2. Credit cards

One of the greatest inventions ever. I don't have to carry wads of cash, with their attendant worry about being searched by the police and having the cash confiscated. Note that I'm not worried about being robbed - just stopped by the cops. Also, always carry a backup, in your shoe if necessary. Cards can be replaced easily - and your liability for purchases you didn't make is limited - but it takes time and you don't want to be stranded in Nebraska without any means of payment. Well, you don't want to be stranded in Nebraska, period.

And no, I don't carry a balance on the things. I use cash-back credit cards so it's a net gain for me. But this isn't the entry for financial advice, so I'll just leave it with this note: credit cards are not evil; they're tools that can get you in trouble if you use them wrong, and be an immense benefit otherwise.

I have credit cards that work internationally so I don't even have to be too concerned about currency exchange.

3. Laptop

I know people who think that a vacation means getting away from the internet. To me, this is bullshit; the internet is awesome, because it lets introverts like me connect with people without actually having to deal with them. And a dumbphone won't cut it - how the fuck do you write anything of any length on a phone? So I lug my gaming laptop wherever I go. It's heavy. I deal. Then the only trick is finding internet connections, which isn't usually a problem domestically.

Note to self: look into satellite connectivity whilst abroad.

There are other necessities, I suppose, such as medications, and some bags to carry everything in. But those are the three biggies, in no particular order. Oh, and another thing I learned: Always Carry Backups. Batteries, CPAP supplies, the extra credit card I mentioned, your GPS (a map is a sufficient backup for that if you're one of the select few, like me, who know how to freaking use one), etc.

A note about maps and GPS: you hear a lot of technophobic shit about GPS devices. They lead you into lakes. They get you turned around. They're unreliable. Bullshit, all of it. People got lost all the time when using paper maps, and anyone who drives into a lake is a goddamned idiot. You can't follow GPS blindly; that can get you into trouble even if you don't end up soaking. They're technological and scientific marvels, yes, but use your freakin' brain, too.

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