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 6 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 8 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').

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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.

December 27, 2023 at 8:38am

#1061468

Solipstick

Think you're not special?

How Do I Know I’m Not the Only Conscious Being in the Universe?

For that matter, how do you know there's a universe and that you're not making the whole thing up?

It is a central dilemma of human life—more urgent, arguably, than the inevitability of suffering and death.

Which wine goes best with salmon mousse?

I have been brooding and ranting to my students about it for years.

You sound fun at parties.

It surely troubles us more than ever during this plague-ridden era.

Article is from September of 2020, but, let's be fair, there's always a plague.

Philosophers call it the problem of other minds. I prefer to call it the solipsism problem.

I call it Rikki.

Solipsism, technically, is an extreme form of skepticism, at once utterly illogical and irrefutable. It holds that you are the only conscious being in existence.

Irrefutable, sure, meaning unfalsifiable (unless we learn something new from this article). Illogical? Nonsense. It's more logical than "we live in a simulation" or "God created the universe in six days, then goofed off."

You experience your own mind every waking second, but you can only infer the existence of other minds through indirect means. Other people seem to possess conscious perceptions, emotions, memories, intentions, just as you do, but you cannot be sure they do.

In numerous cases, I am, in fact, pretty sure they don't.

Natural selection instilled in us the capacity for a so-called theory of mind—a talent for intuiting others’ emotions and intentions. But we have a countertendency to deceive one another and to fear we are being deceived. The ultimate deception would be pretending you are conscious when you are not.

Now, hang on. If others don't exist, they can't be deceiving you. Sure, you can deceive yourself, but that's trivial. And if others do exist, in order to pretend that they're conscious, they have to make the conscious effort to deceive, which means they're conscious, which doesn't necessarily mean they exist...

This is why I went into engineering, not philosophy. Well, that, and the money.

The solipsism problem thwarts efforts to explain consciousness. Scientists and philosophers have proposed countless contradictory hypotheses about what consciousness is and how it arises.

Yes. That's what you do. You propose a hypothesis, then look for evidence against it. Lots of them will turn out to be wrong. Many will turn out to be hard, or impossible, to verify.

Panpsychists contend that all creatures and even inanimate matter—even a single proton!—possess consciousness.

Hard to argue against, but needlessly multiplies entities, so it's probably wrong.

Hard-core materialists insist, conversely (and perversely), that not even humans are all that conscious.

What's perverse about that? It's demonstrably true.

But the solipsism problem is far more than a technical philosophical matter. It is a paranoid but understandable response to the feelings of solitude that lurk within us all. Even if you reject solipsism as an intellectual position, you sense it, emotionally, whenever you feel estranged from others, whenever you confront the awful truth that you can never know—really know—another person, and no one can really know you.

Or you could, I dunno, get over it?

Religion is one response to the solipsism problem. Our ancestors dreamed up a supernatural entity who bears witness to our innermost fears and desires. No matter how lonesome we feel, how alienated from our fellow humans, God is always there watching over us. He sees our souls, our most secret selves, and He loves us anyway. Wouldn’t it be nice to think so?

No.

The arts, too, can be seen as attempts to overcome the solipsism problem. The artist, musician, poet, novelist says, “This is how my life feels” or “This is how life might feel for another person.” They help us imagine what it is like to be a Black woman trying to save her children from slavery or a Jewish ad salesman wandering through Dublin, wondering whether his wife is cheating on him. But to imagine is not to know.

Um, any attempt at communication, however clumsy or brilliant, can be seen that way. Art is, in part, a form of communication.

Love, ideally, gives us the illusion of transcending the solipsism problem. You feel you really know someone, from the inside out, and they know you.

Except that love, in all its forms, is an electrochemical process in the brain. The brain which, perhaps, being all alone in the not-universe, desperately seeks out some evidence that it's not.

For the mentally ill, solipsism can become terrifyingly vivid. Victims of Capgras syndrome think that identical imposters have replaced their loved ones. If you have Cotard’s delusion, also known as walking corpse syndrome, you become convinced that you are dead. A much more common disorder is derealization, which makes everything—you, others, reality as a whole—feel strange, phony, simulated.

Like I said, "this is a simulation" is a mental disorder.

What if those afflicted with these alleged delusions actually see reality clearly?

And what if the bearded guy on the street corner is right?

Conceivably, technology could deliver us from the solipsism problem. Koch proposes that we all get brain implants with Wi-Fi so we can meld minds through a kind of high-tech telepathy.

If you really think about it, though, as a solipsist, that would prove exactly nothing.

Philosopher Colin McGinn suggests a technique that involves “brain splicing,” transferring bits of your brain into mine, and vice versa.

This. This is why philosophers should never be allowed to do science.

But do we really want to escape the prison of our subjective selves? The archnemesis of Star Trek: The Next Generation is the Borg...

Here's the thing, though: Once you're a Borg, once you're assimilated, you're perfectly content. Some might even say ecstatic. Those are brain states, and, like love, are electrochemical processes which can be externally manipulated. It's only from the outside that it looks like something to fight against. Hell, that's what makes them so damn effective as scary bad guys: you know you don't want it, but presumably almost every Borg didn't want it, either, but they've achieved peace in their collective unity.

You know what that sounds like to me? Death. And indeed, it parallels religious concepts of death. Perhaps that's why the Borg were so effective as villains.

In any case, these things can be fun to think about, but, as the author notes, solipsism ends up leading to dark places. And since there are alternatives, why not at least assume not-solipsism?

This is you, urging you to do exactly that.

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