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:

Turkey DrumStik

about 7 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

November 7, 2018 at 12:47am

#945072

Colorblind

I have a friend who is colorblind.

I don't mean that in the sense of he pretends to not be racist; I mean that he has the textbook male red-green colorblindness. He gets by just fine, of course; traffic lights always have red on top and green on the bottom. Except in Atlantic City, where the stoplights are sideways - it's a good thing I was driving, then. But it has led to some amusing stories, such as when he described a gray cat as "green."

When I was a kid, I had a realization: we have no way of knowing if the color I see as green is the same color as what you see as green, and vice versa. If neither of us is colorblind, we can agree that something is green, but if I could somehow see through your eyes, I might interpret it as yellow or red or some color that I've never seen. This was kind of a throwaway realization for me, but decades later, I saw a post where someone presented this as an epiphany they had while stoned, and the typical response was "...whoa." As if it wasn't something I didn't figure out when I was seven years old and most definitely not stoned.

I really should publish my epiphanies more often. Here's one: Creation and destruction are the same thing. Not just two sides of a coin, but the same thing. The only reason we call something one or the other is the value we place on the conditions before and after an event. For instance, consider a tree. You chop it down, destroying the tree - but creating firewood. You burn the firewood, destroying it - but creating warmth and light. Which of the two states you focus on depends only on your perceptions at the time. An argument can be made that creation is locally reversed entropy, while destruction is locally accelerated entropy, and I'd accept that argument to a point, but what about when you destroy a rock to create gravel? If you need the gravel more than you needed the rock, it's creation, even though you've increased entropy in the material.

We can't know if my green is your green, though. Maybe someday the technology will exist to allow us to see through another's eyes and hijack their perceptions, but even then, I'd still wonder. It's an untestable hypothesis, for now. But in at least one sense, we know that your green is not my green - because colors also have emotional associations. I think of money, because money is of great importance to me. Maybe you think of leaves and plants. Someone else might think of recycling and environmental issues. Yet another person could associate it with envy; another, with sickness, or mold, or Kermit the Frog. Maybe even all of these things, and more, at different times, in different moods. Possibly the earliest association I have with green was when I was riding in a car with my parents: "Green means go," they'd tell me, which would serve me well when I got my driver's license. Thanks, mom and dad; I never would have figured that out without your wisdom.

But I've still never seen a green cat.

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