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 5, 2019 at 12:04am

#967271

Quack

I swear not all of the links in my blog fodder queue are from a certain British rag.

But this one is.

https://www.theguardian.com/commentisfree/2019/feb/14/my-patient-swapped-chemoth...

My patient swapped chemotherapy for essential oils. Arguing is a fool’s errand

These days whenever I see "essential oils" my mind replaces that phrase with "snake oil."

Oh, sure, essential oils have their uses. For example, some of them smell nice.

“Tell me why I should have your chemotherapy when I can be healed naturally!”

Shall I reiterate my various rants about the word "natural" (and its adverbial declension)? No? Okay.

I used to think that these second opinions were illuminating for patients and nudged them towards change. But what I have learnt in the last few years is that cancer patients in search of alternative cures are more deeply entrenched than ever in their beliefs. Thanks to the rise of social media, the ability to filter out conflicting viewpoints and a bevy of supporters for every outrageous idea, these people arrive convinced about their theories.

If I could change one thing about our language, it would be to stop using the word "theory" (and its various declensions) to mean "crackpot propaganda."

A survey commissioned by the American Society of Clinical Oncology spoke to more than 4,000 American adults, a quarter of whom were current or former cancer patients. Nearly 40% of those surveyed “somewhat” or “strongly” agreed that cancer can be cured via oxygen, diet and herbs alone.

Again, this article is from a UK source, and it's clear from the discussion of "taxes paying for healthcare" that the anecdote does not take place in the US. But I would be surprised if the level of idiocy in the UK was much different than in that survey. After all, 50% of them or so voted against their own interests on the Brexit referendum (I'm leaving that deliberately ambiguous).

Enzymes, waves and magnets do not cure cancer, and they cost the patient every step of the way. Small bottles of unknown and frequently adulterated or plainly toxic substances cost hundreds of dollars, not to mention every consultation that pretends to read the eyes and sense the energy to cure cancer, even as the patient worsens. How do I know? Because dying patients relate these stories in a last attempt to prevent their fellow patients being duped.

"Dollars?" *checks author bio* Oh, she's from Oz. My comment above about the level of idiocy remains; just ignore the jibe about Brexit.

It has been received wisdom that oncologists can see off quackery through good communication but I’m afraid that isn’t so.

That's because of cognitive bias, something that this author doesn't seem to be affected by; good on her.

Oncologists have been properly entangled in a web of fake news. Their authority has been undermined and their expertise ridiculed by a determined, global and hard-to-track battalion of quacks and their acolytes. Greater vigilance, stronger regulation and improved health literacy might help, but the pull of alternative cures is strong.

The two greatest forces in the universe are compound interest and marketing. Marketing can convince anyone of anything. Hell, it convinced people that Budweiser is beer, Smart Cars are cute, and bottled water should be a thing. I'm not immune, though I'd like to think I have some inoculation against it - but that might be wishful thinking.

Philosophical question: Consider a purveyor of quackery. Either they actually believe in their product, or they don't. In the former case, they are deluded. In the latter, they are committing fraud. The philosophical question is: which is worse?

I'm not sure either is worse. Both are bad. I sometimes wish I had fewer scruples so I could market my cats' shit as a healing balm.

Thing is, though, I have a different perspective than this author. She's an oncologist, who presumably went to medical school and had medical ethics drilled into her. Also presumably, she got into medicine out of a desire to help people - that's reflected in the writing. So her perspective is, "I want to help these patients see the truth so they don't die."

My take is, "fuck them." The more people die from stupidity, the less stupidity there is in the world.

I know that makes me a terrible person, but at least I'm not selling extract of belladonna as a cancer cure.

2 Like • Comment • View 8 comments

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