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 27, 2019 at 12:17am

#961598

Change

From South Point, we head north to visit the volcano responsible for building the island. Kilauea has been continuously erupting for the past 36 years. We board a small boat and motor to the place where the lava spills into the ocean, boiling the water and erupting steam into the air. The lava flows are slow and deceptively innocent, but their power to destroy is great and we are careful not to get too close. Our guide reminds us, however, that equal to the lava’s power to destroy is its power to create.

In your blog today, share something unexpected. Have you been surprised by someone or something recently?

No.

Well, sort of. Speaking of Hawai'i. I mentioned my friend on Maui in yesterday's entry. Well, he's not there anymore.

I mean, it was never going to be a permanent gig for him, but he left earlier than he'd originally planned. Upside: more money. Downside: Not Maui.

It's unfortunate, because I was kind of hoping I'd get to visit there one more time, but alas, if I do, I'm going to have to pay tourist prices everywhere.

So, a bit surprising, but nothing earth-shattering like, say, a volcano.

I've observed an erupting volcano, as well, but not in Hawai'i. The one I saw was on Montserrat, in the Caribbean. No lava flows; lots of ash. By the time I saw it, it had been spewing ash for several years, ash that had buried both the island's airport and its primary town, each on opposite sides of the cone. The population of the island had been reduced from something like 13,000 to closer to 3,000, with the refugees mostly having been relocated to the US and the UK, with some going to nearby islands.

"Equal to the lava’s power to destroy is its power to create." One of the most important realizations I've ever come to was the understanding that creation and destruction are really the same thing. Not opposites; not "two sides of the same coin," but the same thing, depending on how we look at it. Someone tosses a coin, and you call heads or tails, right? You desire a particular outcome: heads, or tails. Then the coin comes up one or the other, which is either the one you wanted or the one you didn't want. Whether we call something "creation" or "destruction" depends entirely on the value we put on one or the other.

You're cold, so you want a fire. So you burn some firewood. That destroys the wood, but at that moment, what you value more is the heat of the flame. To make the firewood, you, or someone, had to destroy a log. That log could, perhaps, have been used for fenceposts or a house; but at that moment, the more valuable use was firewood. To make the log, one had to destroy a tree, which was providing shade, oxygen, maybe a home for creatures; if the tree was already dead, it was still contributing to the ecosystem. And so on.

Drop a nuclear bomb on something, and you destroy it, right? Well, sure, but you are also creating something at the same time. A crater, radiation, a shock wave, etc.

It's a matter of perspective, and perspective alone. Sure, an argument can be made that it might take a year to create, say, a skyscraper, while it only takes minutes to tear it down, but what we call "destruction" can take time as well, so that's not a pure distinction.

Rome wasn't built in a day, or so they say, but neither was the Roman Empire fractured in a 24-hour period.

Sure, you can get into the philosophical argument (backed up to some extent by science) that nothing is created or destroyed, just transformed, but that's not my point - form can be created or destroyed. Or, as I'd argue, both at the same time.

Hawai'i - the Big Island - is the newest of that particular chain, but it won't always be. The Earth's crust there is sliding over a geological hot spot. A while back, that hotspot created Haleakala and the rest of Maui. Before that, it built Oahu. And down the line, to Kauai and beyond. And right now, off the coast of the big island, the next Hawaiian island is being formed.

Even on geological timescales, change is the only constant.

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