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 15 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 16 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.

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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.

April 27, 2020 at 12:03am

#982160

Fearful Symmetry

Today's link is just some really cool theoretical physics.

https://www.quantamagazine.org/how-simple-rules-bootstrap-the-laws-of-physics-20...

Why the Laws of Physics Are Inevitable
By considering simple symmetries, physicists working on the “bootstrap” can rediscover the basic form of the known forces that shape the universe.

Fear not; there's no advanced math in here. This is mostly of philosophical interest. One of the questions I keep seeing in physics is, "Why is there something rather than nothing?" Though that seems to me that it might be asking the wrong question; after all, if there were nothing, we wouldn't be around to ask the question.

To an astonishing degree, nature is the way it is because it couldn’t be any different.

That's probably the answer to the question that they didn't ask. For a long time, I've seen speculation that other universes might exist, somewhere, with different laws of physics. This seems to suggest that can't be the case.

Since the 1960s, and increasingly in the past decade, physicists like Baumann have used a technique known as the “bootstrap” to infer what the laws of nature must be.

It's satisfying when you can come at a question from different angles and get the same result.

That solution is the graviton: a spin-2 particle that couples to itself and all other particles with equal strength.

I'm no expert on these things, but my understanding is that, so far, the graviton is a purely theoretical particle. Physicists have been proposing it for some time, because it would imply quantum gravity, which in theory could bridge general relativity and quantum mechanics, perhaps leading to a Grand Unified Theory, kind of the Holy Grail of physics.

Of course, assuming they find this Holy Grail ("I told him we already got one"), it'll probably just raise more questions. This is why science is fun.

It’s “just aesthetically pleasing,” Baumann said, “that the laws are inevitable — that there is some inevitability of the laws of physics that can be summarized by a short handful of principles that then lead to building blocks that then build up the macroscopic world.”

Complexity emerging out of simplicity just makes so much more sense than the other way around, don't you think?

2 Like • Comment

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