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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 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 10 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.

March 5, 2024 at 11:12am

#1065605

Ignorance is...

Here's an example of finding an article that aligns with my existing worldview but having my doubts.

New Study Links Optimism To Lower Cognitive Abilities

That doesn't mean it's right, of course. But it also doesn't mean it's wrong.

Optimistic thinking, often celebrated in self-help literature as a pathway to happiness, health, and longer life, can also lead to poor decision-making.

This has been clear to me for a while.

Research from the University of Bath shows that excessive optimism is actually associated with lower cognitive skills such as verbal fluency, fluid reasoning, numerical reasoning, and memory. Whereas those high in cognitive ability tend to be both more realistic and pessimistic in their expectations about the future.

Basically: ignorance is bliss.

Well, no, not really, because ignorance is simply not knowing something, which has nothing to do with how smart you are. Everybody's ignorant about lots of things. But stubborn, willful ignorance is another matter entirely... also not necessarily correlated with intelligence.

In any case, I'll note that the passage I just quoted seems to be very careful not to imply causation. Does excessive optimism make a person less clever, or does being an idiot lead to optimism? Or, alternatively, is there something else causing both effects? Much as I want to believe the article, that seems like an important thing to find out.

“This points to the idea that whilst humans may be primed by evolution to expect the best, those high in cognitive ability are more able to override this automatic response when it comes to important decisions. Plans based on overly optimistic beliefs make for poor decisions and are bound to deliver worse outcomes than would realistic beliefs,” Dr Dawson added.

Okay, I'm going to need a reference on that "primed by evolution" thing, hopefully not from bogus evolutionary psychology. Also, why isn't this "override" attributed to evolution?

“Unrealistically optimistic financial expectations can lead to excessive levels of consumption and debt, as well as insufficient savings. It can also lead to excessive business entries and subsequent failures. The chances of starting a successful business are tiny, but optimists always think they have a shot and will start businesses destined to fail,” Dr Dawson said.

But, clearly, they're not all destined to fail (unless you take the really long view that every business will fail eventually, even if it takes a few centuries). If we didn't have people taking a shot at these things, we wouldn't have businesses at all. It's like the baby turtles' march to the sea.

The study took data from a UK survey of over 36,000 households and looked at people’s expectations of their financial well-being and compared them with their actual financial outcomes. The research found that those highest on cognitive ability experienced a 22% increase in the probability of “realism” and a 35 percent decrease in the probability of “extreme optimism”.

For once, I can't complain about the sample size. But there are other obvious yellow flags here. UK only, for starters. How cognitive ability was determined from a mere survey: did they basically ask trivia questions? Even if not, there are major issues concerning the standard IQ tests. And how do we know that they're studying truly independent variables?

“Unrealistic optimism is one of the most pervasive human traits and research has shown people consistently underestimate the negative and accentuate the positive. The concept of ‘positive thinking’ is almost unquestioningly embedded in our culture – and it would be healthy to revisit that belief,” Dr Dawson added

Sounds to me like someone hated the idea of positive thinking and set out to debunk it. I can understand that impulse, as I share it. But, first, if science goes into an investigation wanting a certain outcome, that outcome is more likely (even excluding fraud or other shenanigans). And second, I have serious questions about the methodology.

So I put this in the category of "stuff I want to believe, and which tracks with stuff I've been saying, but can't trust the science."

3 Like • Comment • View 1 comment

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