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 8 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 10 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.

January 22, 2024 at 10:29am

#1062791

For Short Values of 'Lifetime'

A little perspective, from Cracked...

5 ‘Once in a Lifetime’ Events That Are Actually A Lot More Frequent Than Once in a Lifetime

I'll just jump into the list, which, naturally, counts down.

5. Catching A Foul Ball

Sure, it’s no home run ball, but catching a foul ball while watching a live baseball game is still a nice consolation prize.

Judging from some of the videos I've seen, there's more competition, with fewer rules, between the spectators near the end of the ball's trajectory than there is on the field.

The fact is, though, that if you’re a frequent attendee, especially if you’re picking seats based on likely trajectories, the odds really aren’t that astronomical. They basically sit right around 1 in 580.

Hell of an "if" there. I haven't been to a baseball game as a spectator since Nixon was president. Hell, I've played it more than I've watched it. I guarantee you me catching a foul ball would not only be a once-in-a-lifetime experience, but a probability-breaking macroquantum event.

4. Total Solar Eclipse

Would it surprise you I knew about this one? No? Okay.

It’s true. The belief that it’s so incredibly rare is probably based on the fact that it takes 375 years to happen again in the same location. If you’ve got a deep love for orbs and an even deeper bank account, though, you could see dozens in your lifetime with the right travel plans.

Yeah, but let's not confuse "happens somewhere on Earth" with "happens where you live." Tell you what, though, if you live around Paducah, Kentucky, you experienced totality in 2017, and will do so again this April. So you're, you know. Doomed.

3. Animals Going Extinct

The fact is, animals are going extinct constantly, in huge numbers. The reason we don’t see more coverage of each particular critter configuration we’re losing isn’t because it’s not happening, it’s because there are literally too many to report on without their own 24-hour news cycle.

2. Blue Moons

I'll give them points for not explicitly stating the false definition here. But they did include a link to a site that still promulgates that mistake, so negative points for that. By the false definition, a few years ago, we had one in January and one in March. That can never happen with the true definition.

The actual last Blue Moon was on August 22, 2021. The next one is this year, on August 19. After that, May 20, 2027. One can only occur in August, November, February, or May. But yeah, they happen every 2-3 years. Which is rare enough to warrant the "once in a blue moon" expression, but the false Blue Moon can never, ever coincide with the true Blue Moon.

To mash up two entries into one fun fact to throw into bar conversation, “once every total solar eclipse” would actually be more frequent than “once in a blue moon.”

Again, sure, if you're privileged enough to be able to travel to the eclipse.

1. Investment Opportunities

And this one should go without saying.

If someone wants you to buy what they're selling, it's because they think your money is worth more than what they're selling. You should probably think so, too. Which should be obvious, but greed can blind you.

3 Like • Comment • View 1 comment

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