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 11, 2024 at 11:06am

#1062201

Seeing is Disbelieving

The Random Number Gods have chosen to bless us with another Cracked article this day.

4 Illusions That Trick You in Your Everyday Life

And no, one of them isn't "time." Well, the last one is. Sort of. Not really.

Don’t believe your eyes. They lie to you every day.

I might have mentioned this before, but I used to work with a guy who was constantly showing me optical illusions. As he was an engineer with more experience than I had, I thought he was trying to train me to believe the measurements and numbers more than my visual perception. But no; he was also a devout Christian and was trying to show me that only God is real or whatever.

Thing about those optical illusions you can find all over the internet is that most of them have documented, logical, proven explanations. So all that did was reinforce my admiration for science.

4. Your Nose Is Smaller Than Selfies Tell You It Is

Long ago, I was a semi-professional photographer (in that I got paid for it, but it wasn't my main source of income). This section reiterates what I've known for a long time about the difference between focal lengths.

For authentic pics, never hold your phone up close. Instead, set up a tripod several feet away, inconveniencing everyone in the name of better art.

Like anyone's gonna do that. Just hand your phone to some random stranger and ask them to take the pic. There's a very good chance you'll even get your phone back, afterwards.

3. The Moon Keeps Changing Its Size on You

Yes, this one was the real reason I saved this article to share.

If you’re lucky enough to see the Moon rise or set over the horizon, you’ll notice it looks different from how it looks when it’s high in the sky. It looks a bit more yellow, due to the greater amount of atmosphere that moonlight must travel through to reach your eyes from this angle. It may also look slightly squashed, which is another effect of looking at it through so much atmosphere.

I should note that there are times when the Moon really is bigger or smaller because its orbit, like pretty much every orbit, is elliptical. This horizon illusion dwarfs that effect, though, at least in my experience.

More than anything else, however, the low Moon simply looks much bigger than the Moon as you normally know it. Everyone sees this difference, and no one knows why.

Perhaps, but I'm convinced it's a combination of things. Including a version of the Ponzo Illusion

(not to be confused with the Ponzi Scheme). Flat Earth (not really, but we perceive it as such), mostly flat sky converging on it. Also the convergence of the two horizon lines. I'm sticking with that hypothesis until smarter people than me figure something else out. That might take a while.

2. They’re Using Color to Lie About Produce

Ever since that photo known as The Dress exploded into public consciousness, I've seen more and more articles explaining color perception. I got really tired of seeing it when it first happened, but apparently, some good has come from it in terms of public perception of color perception.

1. Clocks Look Like They Stop Because They Break Your Brain

Despite what I said above, no, time is not an illusion (probably). As with color, though, our perception of time can change with circumstances.

The article goes into the science here, and since it tracks with stuff I already knew, it's absolutely worth reading. Spolier: it's not a glitch in the Matrix. There is no Matrix, except in the movies.

In conclusion, my mom always told me "seeing is believing." Sorry, Mom. You were wrong.

2 Like • Comment • View 2 comments

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