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!

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 3, 2024 at 11:04am

#1061794

Wilson!

I saved the link to today's article because it's about a place I've been to fairly recently; and, of course, because it's about astronomy.

Saving Mt. Wilson Observatory: Inside the long battle to maintain the spot where we found our place in the universe

Gotta say, actually seeing that telescope was inspiring, even if I did have to climb up flights of stairs in the rarefied mountain air to do it.

Los Angeles was once the best place in the world to see the universe.

Now, a lesser person might retort that Los Angeles is still the best place in the world to see stars, but I'd never stoop so low. Never, ever.

The most important things we know about the cosmos were discovered in the early 20th century at Mt. Wilson Observatory.

Mostly, that not only is space mind-bogglingly big, but it's getting bigger.

It was here, 100 years ago, that Edwin Hubble noted a light in the distance that would lead to one of science’s greatest discoveries.

Other sources say 1924 was the year, but this led me to check the publication date: "Oct. 3, 2023 3 AM PT." Someone was keeping astronomer's hours.

And Mt. Wilson Observatory discovered what so many aging luminaries in L.A. have: Once your starlight fades, no one is quite sure what to do with you.

See, it's not just me who won't admit to thinking of these things.

With an annual budget smaller than that of some fancy L.A. parties, a nonprofit organization and volunteers have done a heroic job of keeping the grounds and telescopes open for visitors and the few scientists still working there.

And I'm glad they do. Even if you're not a huge astronomy nerd like I am, the place is worth a visit for the scenery and views.

The observatory sits at the summit of 5,715-foot Mt. Wilson, accessible only by a serpentine stretch of Angeles Crest Highway.

It's not that serpentine, compared to some other roads I've been on in California.

In 1908 the animals delivered a 60-inch wide, 1,900-pound mirror to collect light from the stars — the keystone of the largest and most powerful instrument of its kind in the world.

And that wasn't even the final telescope mirror. The one Hubble used was 100 inches across. Based on the photo in the article, that one was delivered not by mule, but by some newfangled internal-combustion vehicle.

So, just for comparison: 100 inches is a bit over 2.5m. The famous space telescope which is the only reason non-nerds have even heard of Edwin Hubble "only" has a 2.4 meter mirror, but it definitely wasn't delivered by mules... being, you know, in space and all. (The "being in space" thing was the real advancement; even discounting L.A.'s various forms of pollution, it helps to not have atmosphere in the way.)

The human eye only collects up to 0.2 seconds of visual data before uploading it to the brain. A camera can collect light over a much longer period, which is why long-exposure photographs of the night sky contain far more stars than are visible with the naked eye.

An astronomer once pointed out to me that the popular conception of telescopes as magnifying devices isn't really correct. They're actually light-gathering devices.

Quite a bit of the article goes into the bio of Hubble and the history of his universe-altering (well, actually, his "our conception of the universe"-altering) discovery.

Despite its august place in scientific history, Mt. Wilson was no longer the best place on Earth to do astronomy. Carnegie pulled its funding from the observatory and closed the big telescope in June 1985.

Honestly, I'm surprised it lasted that long.

There are actually several observatories (or former observatories) in the L.A. area. The most famous, and picturesque, isn't Mt. Wilson, but Griffith. It's located close enough to Hollywood that it features in many establishing shots of that area. It's a museum (and park) now, but it's an awesome museum. The article mentions it, along with efforts to keep Mt. Wilson open.

As noted, science is mostly done elsewhere, now. Maintaining Mt. Wilson isn't about that; it's about recognizing its place in history. Everything's subject to entropy. The famed Hubble Space Telescope will eventually (like within a decade) de-orbit, and all we'll have left of it is a few stunning photographs, maybe some debris, and, of course, its additions to the ever-changing course of science. But this one, we could keep... if we have the will.

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