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.

March 21, 2019 at 12:47am

#954673

Depressing

https://aeon.co/essays/the-evidence-in-favour-of-antidepressants-is-terribly-fla...

I've ranted here about certain science issues, though specifically I've noted my problems with nutrtitional studies and the way they're reported. This one, however, is about antidepressants.

Do antidepressants work?
Depression is a very complex disorder and we simply have no good evidence that antidepressants help sufferers to improve

Like many people, including some I consider role models or otherwise greatly admire, I've struggled with depression. I know there are readers here with mental health issues as well. Let me state up front that I've never been suicidal - that's always the first question people ask, so I wanted to get that out of the way.

As always, let's first consider the source. I have no problem with Aeon, but I do wonder about the particular author:

Jacob Stegenga is a lecturer in philosophy of science at the University of Cambridge. He is the author of Medical Nihilism (2018). He lives in Cambridge.

Well, since I ragged on Oxford the other day, it's only fair to consider Cambridge. When I see the words "philosophy of science," my inner skeptic goes on high alert. In fact, anytime I see "philosophy," this happens. I'm not saying philosophy is bad - I think it's important - but it's very easy to draw utterly logical conclusions from bad premises. So one needs to be aware of any lurking biases - ironic, because the article discusses biases in studies. Also, this might be an ad for the guy's book. Not that there's anything wrong with advertising your book.

Your grief and guilt overwhelm you. You are so tired you cannot think straight. Your simple joys are lost in an invisible agony. You have pain in your head and back and stomach, real pain. The swamp of your soul suffocates you with despair. All this is your fault, you are worthless, and you might as well die. This is how depression can feel, though people’s experiences of it, including the severity of symptoms, can vary widely. This terrible disease affects about one person in 10 at some point in life and, to treat it, many millions of people have taken antidepressants. Unfortunately, we now have good reasons to think that antidepressants are not effective.

Okay, gotta be honest here - my own experience has never been that severe. I've been prescribed antidepressants in the past, mostly SSRIs, some SNRIs, and if you've had any experience with this issue yourself, you probably know what those are. Those medications act on the neurological pathways that affect certain neurotransmitters in the brain. It's pretty basic stuff as far as biological psychology goes, but that field is itself not what I'd call "basic."

The meds each did one of three things for me:

1) Nothing.
2) Hallucinations, and not fun ones.
3) Made things worse.

To be fair, (3) might have happened anyway - can't tell with a sample size of one. I'm pretty sure that (2) isn't what was supposed to happen, and (1) happens more often than we'd like. And that's kind of the point of the article.

The vast majority of these studies are funded and controlled by the manufacturers of antidepressants, which is an obvious conflict of interest.

And now we get into some crossover with my problems with nutritional science.

The trials that generate evidence seeming to support antidepressants get published, while trials that generate evidence suggesting that antidepressants are ineffective often remain unpublished (this widespread phenomenon is called ‘publication bias’). To give one prominent example, in 2012 the UK pharmaceutical company GlaxoSmithKline pleaded guilty to criminal charges for promoting the use of its antidepressant Paxil in children (there was no evidence that it was effective in children), and for misreporting trial data.

And Paxil was the shit that made me hallucinate. Look, if you want me to hallucinate, give me LSD. Not that I've ever tried it, but I've heard that microdoses might be effective in treating depression - moreso than antidepressants, anyway.

The placebo effect is when patients improve merely as a result of the medical care they have received rather than as a result of the biochemical properties of their drug. The idea is that the mere expectation that you will get better after receiving medical care can itself contribute to you getting better. Some diseases are more responsive to placebo than others, and depression is one of the most placebo-responsive of all diseases.

This makes sense, though entire articles are out there that go into more detail about the placebo effect. Unfortunately, the response of depression to placebo is one thing that leads some people to believe that it's "all in your head" and you can just think yourself better. There might be some truth to that, but it doesn't mean the disorder isn't "real." I mean, technically, pain is all in your head - well, your nervous system in general - but only assholes think you can think yourself out of pain.

In short, we have plenty of reasons to think that antidepressants have no clinically meaningful benefits for those suffering from depression. Conversely, we know that these drugs cause many harmful side-effects, including weight gain, sexual problems, fatigue and insomnia. Some studies have demonstrated a link between antidepressants and the risk of violence, suicide, childhood and teenage aggression, and psychotic events in women.

Which would fall into my "making things worse" category above.

Conversely, there is another theoretical consideration that seems to speak against antidepressants. Some critics claim that many diagnosed cases of depression are not cases of real disease, but rather involve the ‘medicalisation’ of normal life – normal grief, stress, anxiety or simply suburban sadness being brought into the jurisdiction of medicine.

I've had some experience with this, I think. A doctor prescribed me some SNRI medicine because I just couldn't be arsed to cook my own food (healthy) when it was so easy to order from GrubHub or stop by McDonald's (not quite as healthy). After taking the medicine for a while, I still ate at McDonald's. As I told the doctor when I informed her I'd be tapering myself off of them, "they still haven't come up with a pill to fix moral failings."

How, then, did I get past that? Well, I don't know, but antidepressants didn't do it.

So, bottom line? Apparently, if a course of treatment works for you, then it works. It doesn't mean it would work for me. I keep hearing that a positive attitude can make you healthier and live longer. If so, this might well be my last blog entry, because every time I get a positive attitude, I bury it like the abomination that it is.

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