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').

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.

January 5, 2022 at 12:03am

#1024195

Ignorant Bliss

Dropping another bucket into the "JAFBG"

[XGC] well...

Do you think intelligence and happiness are interconnected? Is an intelligent person more or less likely to be happy?

Obviously they are negatively correlated, because I'm hyperintelligent and grumpy as fuck.

Okay, yes, that's a joke on many levels. Also, one data point doesn't mean anything, and my opinion isn't worth the toilet paper it's smeared on. So, putting aside for the moment my War on Happiness, let's see what the science says.

But before I do, I will make a prediction, and let's see if I'm right. If I am, then you will all have to agree that I'm a supergenius; if I'm wrong, you can call me an ignorant dumbass.

That prediction is: Higher IQ will be associated with lower "happiness" (which as I've noted before isn't a quantifiable thing and is necessarily subjective). Probably because we giant-brains see the world as it is and are utterly appalled by all the willful ignorance and dumbassery, while the willfully ignorant dumbasses achieve brainless bliss.

Here's one. The relationship between happiness and intelligent quotient: the contribution of socio-economic and clinical factors

(PubMed)

Results: Happiness is significantly associated with IQ. Those in the lowest IQ range (70-99) reported the lowest levels of happiness compared with the highest IQ group (120-129). Mediation analysis using the continuous IQ variable found dependency in activities of daily living, income, health and neurotic symptoms were strong mediators of the relationship, as they reduced the association between happiness and IQ by 50%.

Conclusions: Those with lower IQ are less happy than those with higher IQ. Interventions that target modifiable variables such as income (e.g. through enhancing education and employment opportunities) and neurotic symptoms (e.g. through better detection of mental health problems) may improve levels of happiness in the lower IQ groups.

Wait, that can't be right. Besides, the study only goes up to 129, and mine's in the 180 range. I gotta find an article that supports my confirmation bias.

New research suggests that IQ leads to greater well-being by enabling one to acquire the financial and educational means necessary to live a better life

(Scientific American)

Crap, you're not helping here. Now I have to venture into the fetid swamp of Psychology Today.

The Surprising Connection Between Intelligence and Happiness

It turns out that IQ—even assessed in childhood—does predict the emotional ups and downs a person will have over the course of her life. People who were below average in intelligence experienced significantly more variability in their life satisfaction than did those who were above average.

No, no, no, no lalalalalala.

I know, I can usually trust BBC...

IQ linked to levels of happiness

People with lower intelligence are more likely to be unhappy than their brighter colleagues, according to UK researchers.

Goddammit. (Yes, I know that one just quotes one of the same studies.)

Okay, so I'm a grumpy, ignorant dumbass. Now sod off.

On a more serious note, this sort of thing should demonstrate why science is important. You can come to your own conclusions based on lived experience or personal observations, but those are utterly useless for drawing general conclusions. Like, "I've never worn a seatbelt and I'm fine!" or "I got the 'rona and just had the sniffles, so this whole thing is overblown!"

It's also wise to, when presented with evidence that contradicts your previously-held beliefs, don't dig your heels into your previously-held beliefs no matter what. Doing that, you look like a toddler having a meltdown in the cereal aisle.

At the same time, we can't just accept studies at face value all the time. Things like "intelligence" can have a few different definitions, and as with a lot of science, the layperson's definition isn't necessarily the same as the technical definition. For instance, a lot of people think that intelligence means being able to remember lists of facts, basically what's needed to win a trivia contest. That's not intelligence, though; that's memory. It may be correlated, but intelligence is way more than remembering shit. Also, again, happiness isn't really measurable and is subject to different interpretations.

Your lived experience is important when it comes to storytelling, but it's otherwise completely meaningless, and the sooner you accept that, the happier you'll be.

But what do I know? I'm an idiot.

In conclusion, you really need to look at this graph

, but I'm not taking up space in my portfolio to make it an embedded image. Just click on the link.

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