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 2 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 4 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.

August 5, 2021 at 12:05am

#1015165

I Drink Therefore I Am

Today we're thinking about thinking. Or we could think about thinking about thinking, in indefinite regression, which I do when I'm feeling particularly lazy.

That Is Not How Your Brain Works
Forget these scientific myths to better understand your brain and yourself.

Another scientific overview, this one purporting to dispel certain outdated information. It's dated March 3 of this year, so maybe some of its information is itself out of date.

The 21st century is a time of great scientific discovery.

And not all the discoveries are fun ones.

Cars are driving themselves.

That's technology, not science.

Vaccines against deadly new viruses are created in less than a year.

And half the people refuse to take them, some because they were created in less than a year. (They weren't, actually; mRNA vaccines in particular built on prior years of research.)

The latest Mars Rover is hunting for signs of alien life.

Even the words "the latest Mars Rover" are pretty cool when put like that.

But we’re also surrounded with scientific myths: outdated beliefs that make their way regularly into news stories.

And sometimes, outright misinformation.

As a neuroscientist, I see scientific myths about the brain repeated regularly in the media and corners of academic research. Three of them, in particular, stand out for correction.

I guarantee you there are more than three of them, but I suppose every article has its word limit.

After all, each of us has a brain...

With the possible exception of anti-vax cretins.

...so it’s critical to understand how that three-pound blob between your ears works.

Or, in the case of anti-vaxxers, doesn't.

Myth number one is that specific parts of the human brain have specific psychological jobs.

I'm just going to highlight from here on out; the article obviously goes into more depth.

This view of the brain became popular in the 19th century, when it was called phrenology. Its practitioners believed they could discern your personality by measuring bumps on your skull.

"Hm. This person is a klutz, and this other person likes to be in bar fights."

Most neurons have multiple jobs, not a single psychological purpose.

Main job, secondary job, side hustle. The economy sucks.

Perhaps the most famous example of puzzle-piece thinking is the “triune brain”: the idea that the human brain evolved in three layers. The deepest layer, known as the lizard brain and allegedly inherited from reptile ancestors, is said to house our instincts. The middle layer, called the limbic system, allegedly contains emotions inherited from ancient mammals. And the topmost layer, called the neocortex, is said to be uniquely human—like icing on an already baked cake—and supposedly lets us regulate our brutish emotions and instincts.

"Supposedly."

And if a uniquely human and rational neocortex controls those beasts, then we have the most highly evolved brain in the animal kingdom. Yay for humans, right? But it’s all a myth. In reality, each species has brains that are uniquely and effectively adapted to their environments, and no animal brain is “more evolved” than any other.

Which is what I've been saying with my highly-evolved brain.

Myth number two is that your brain reacts to events in the world.

Well, first we have to know about said events, and sometimes those are filtered through questionable sources, so... oh, wait, she's talking about something else.

All your neurons are firing at various rates all the time. What are they doing? Busily making predictions.

No matter how hard we wish it were true, we are not living in the present.

The third myth is that there’s a clear dividing line between diseases of the body, such as cardiovascular disease, and diseases of the mind, such as depression.

Yeah, I've known this for a while. One semester of biological psychology doesn't make me an expert on neurology, but it is enough to understand that mental illness is often a form of physical illness.

The idea that body and mind are separate was popularized by the philosopher René Descartes in the 17th century (known as Cartesian dualism) and it’s still around today, including in the practice of medicine.

(Hence today's title.)

Every mental experience has physical causes, and physical changes in your body often have mental consequences, thanks to your predicting brain.

One interesting recent line of research, which I don't think the article mentions, is how intestinal microorganisms can affect one's mental health. And it's trivially true that injuries to the brain can cause mental illness. The brain is, after all, a physical organ, and it can get off-track just like any other organ.

Under the hood, however, your brain creates your mind while it regulates the systems of your body. That means the regulation of your body is itself part of your mind.

Mind (and liver and spleen and gonads) blown.

We adapt to what we learn. But sometimes hypotheses are so strong that they resist change. They are maintained not by evidence but by ideology. They become scientific myths.

That's the real takeaway here, at least in my opinion: our body of knowledge changes, and we need to adapt to it. It can be wrong, but it's still the best means we have of understanding the world outside... and the one within. When you hear some scientist say "A is true," and then a couple of months later say, "A is not true; B is true," they're not being wishy-washy or flippy-floppy; they're adapting to new data.

It would be good if more people could do that.

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