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 an hour 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 3 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.

November 10, 2023 at 10:51am

#1059254

Area 51

This one's from Atlas Obscura, and will surely put to rest the pervasive rumors about secret government alien conspiracies.

Area 51, Aliens, and the Truth (It’s Out There)
A national security historian explains what’s really going on at the infamous Nevada site.

Ah... but that's what they want you to think.

One of the reasons people can never be entirely sure about what is going on at Area 51 is that it is a highly classified secret military facility.

Someone is entirely sure about what's going on. They're not talking.

It was not until 2013 that the United States government even acknowledged the existence and name “Area 51.”

After which they heaved a great sigh and opened Area 52, because Area 51 wasn't secret enough anymore.

As a national security historian, I know there’s a long history of secrets at Area 51. I also know that none of those secrets have anything to do with space aliens.

Or is the author lying?

The base commonly referred to as Area 51 is located in a remote area of southern Nevada, roughly 100 miles (161 kilometers) from Las Vegas.

It's also immediately adjacent to where they did underground and open-air nuclear bomb tests. Perhaps the aliens require a higher level of ionizing radiation to survive.

Area 51 was selected in 1955 to test the U-2 in part because its remote location could help keep the plane secret.

Where the streets have no name...

Area 51 became the test site for other secret new aircraft. This included the A-12, which, like the U-2, was a fast-flying reconnaissance plane. The A-12 was first test flown at Homey Airport in 1962. It had a bulging disc-like center to carry additional fuel. Its shape and shiny titanium body could well have been responsible for some people’s reports about seeing spherical ships, also known as flying saucers.

Why? Were they made out of swamp gas?

When the government does not tell the public the full truth, no matter the reasons, secrets can lead to wild speculation. Secrecy can leave room for conspiracy theories to develop.

And when the government does tell the truth, conspiracy theorists assume it's lying.

In fact, many UFO sightings match almost exactly with dates and times of flights of then-classified experimental aircraft. We also know that prototype drones and more recent versions have been tested at the site.

"Many." So... even if you're telling the truth, others are still aliens?

In the end, there is no reason to think that anything other than earthly technologies have been behind the strange sights and sounds at Area 51.

Ah... but that's what they want you to think.

Okay, seriously (everything I wrote above should be taken with a grain of iodized salt), the government needs to test their super-secret spy technologies somewhere. The only place in the US more remote than the middle of freakin' Nevada would be in Alaska, and that presents logistical challenges, such as either sending stuff across Canada or over the ocean, not to mention it's cold. Though I'd wager there's secret shit going on in that state, too. There's secret shit going on in a lot of places, including a site not far from where I live.

One day, it'll all come out in the open... and then certain people still won't believe it.

6 Like • Comment • View 5 comments

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