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').

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.

July 5, 2019 at 12:17am

#962065

Matter of Fact It's All Dark

What did you dream about doing as an adult when you were little? Have you been able to achieve any of your childhood dreams? Share a fact about your life now that you never imagined would come true.

Hell if I remember.

Funny word, "dreams." It can mean those movies that play in your head when you're asleep, or it can mean a more conscious imagining of what could come to you. You can usually figure out the meaning from context. MLKJr wasn't talking about nocturnal Technicolor when he presented his famous "I have a dream" speech; I don't know exactly how we know that, but we do.

My earliest conscious memory is of being called into another room to watch the first moon landing. I probably had memories before that, but they too have been lost to time. Memories are like that; if you don't constantly refresh them, they dissipate. It's the same way with dreams - both kinds. And even if you do constantly refresh them, they transform, changing into something else. The past becomes its own dream.

When the landing happened - fifty years ago this month, as we're being reminded every time we fire up the internet - I had no reference point, no idea of its significance, no context. It was only later that I became aware, somehow, of the meaning of that watershed moment in human history.

I'm sure I'll have plenty more to say about that, later this month, just like everyone else in the fucking world.

As I've said before, my dad was a sailor by profession. At the time in question, he was still on active duty in the Coast Guard, though at some land-based desk job; he would retire a year later. So I don't even remember if he was there with me, then. It was my mother who made sure I saw the grainy footage, received live, with only the inescapable delay resulting from the vast distances involved, on a black and white television getting its signal from an aerial antenna.

When Dad was home, which was all the time after he retired, he'd pick the coldest, clearest nights to wrap tiny me up in a blanket, carry me outside, and point out the stars - beacons with which he was intimately familiar, because he came from a time when they were crucial to ocean navigation. He had a sailor's understanding of astronomy: constellations, the names of the brightest stars, how they can help you find your way when you're surrounded by vast emptiness.

He always pronounced Betelgeuse incorrectly. I think of him every time I see Orion shining in the winter sky.

Years later, he showed me a photograph he took at some point during his career. If I remember correctly - and I probably don't - he was on a ship somewhere south of the equator when a solar eclipse occurred. Merchant Marine, not Coast Guard. I don't know if they deliberately sailed into the path just to see the eclipse; I like to think they did. I remember thinking, and perhaps saying, that I wanted to see a solar eclipse one day.

I didn't get a chance until two years ago. Orbital mechanics and budget never really cooperated until then. It was easily the coolest thing I ever experienced. No photograph can do it justice, though with modern digital technology, you can get much closer than that old silver-halide image my father had. I didn't bother taking a picture; three million other people would also be viewing this Great American Eclipse of 2017, and easily two million of them could have taken a better picture (and that might be me overestimating my photography skills). I just wanted to experience the fulfillment of the one dream I could remember from my youth.

I'm now the same age my father was when the "Eagle" landed (don't bother doing the math; I was adopted late in my parents' lives), and I'm still looking at the stars.

And the moon.

All that you touch
And all that you see
All that you taste
All you feel
And all that you love
And all that you hate
All you distrust
All you save
And all that you give
And all that you deal
And all that you buy
Beg, borrow or steal
And all you create
And all you destroy
And all that you do
And all that you say
And all that you eat
And everyone you meet
And all that you slight
And everyone you fight
And all that is now
And all that is gone
And all that's to come
And everything under the sun is in tune
But the sun is eclipsed by the moon

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