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').

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.

October 24, 2019 at 12:37am

#968339

Time to Get Real

https://www.brainpickings.org/2016/09/19/a-new-refutation-of-time-borges/

A New Refutation of Time: Borges on the Most Paradoxical Dimension of Existence

"Most paradoxical dimension of existence," my ass. I mean, yeah, it's different from the spatial dimensions, but the only "paradox" is that we can't move freely in both directions of time. And, maybe, that it's not discontinuous; there's no "wall" or barrier to time. That's not really a paradox; that's what time is, and how it's distinguished from the other aspects of spacetime.

Sometimes, philosophy and science collide. This is one of those times.

Time, in other words — particularly our experience of it as a continuity of successive moments — is a cognitive illusion rather than an inherent feature of the universe, a construction of human consciousness and perhaps the very hallmark of human consciousness.

Steaming pile of bovine excrement.

Time proceeded to do time stuff before we evolved, and will continue to do so until the heat death of the universe (because, literally, time stops when entropy stops).

Borges begins by noting the deliberate paradox of his title, a contrast to his central thesis that the continuity of time is an illusion, that time exists without succession and each moment contains all eternity, which negates the very notion of “new.”

Yeah... no. I'm not going to get into light cones and other features of relativity, but, with due respect to the philosopher, this "each moment contains all eternity" crap is crap.

Time, Borges notes, is the foundation of our experience of personal identity...

It is the case that personal identity depends on time. That is, we remember the past but not the future; time is asymmetrical. This, again, is a feature of time. But that's what makes it real, not illusory.

Look, I'm not saying the essay excerpted in the article isn't worth reading. It's always good to gain new perspectives (I've heard that weed assists with this effort). But... I know I've said this before, but if time is an illusion, then space, too, is an illusion, along with everything in it. And if everything is an illusion, while the only things that are "real" are constructs of the mind such as "eternity" or "love" or "the present moment," then this switches the definitions of "real" and "illusion."

Deny the existence of objective reality all you want, but you don't get to make up stuff and then call it real; that's the worst kind of hubris. Sure, there's a lot we don't understand, but any philosophical examination of the universe has to start with something like "the chair I'm sitting in is real," or the entire edifice is built on - quite literally - an illusory foundation.

2 Like • Comment

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