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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Brandiwyn🎶

a day ago
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.

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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.

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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').

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

February 11, 2021 at 12:01am

#1004202

Free

Well, the random number gods have frowned upon me and revealed unto me yet another Atlantic article. I suppose I could have ignored the result and picked something else, but I've got a system. Besides, at least it's not on the same topic.

It is possible that you might hit a paywall with this one. There are ways around it if you care.

One Legacy of the Pandemic May Be Less Judgment of the Child-Free
The coronavirus could change lingering cultural assumptions about what makes for a full and happy life.

While the article is six months old, I don't think it's outdated yet.

My friends were getting honest about how hard it is to raise children right now.

People have, in my experience, always been honest about how hard it is to raise children. But they almost always end the discussion with some variation of "But it's worth it." Reading between the lines here, I'm wondering if the author is responding to the absence of the "But it's worth it." But that may be my own bias talking.

I also read it as an indirect plea to not take my child-free privileges for granted.

Life without children can be easy sometimes. But it's worth it.

I’ve always been ambivalent about whether I would have children, but as I entered my early 40s, I started exploring the possibility of having a child on my own. And then the pandemic happened.

I will point out here that any discussion of whether or not to become a parent is different depending on the gender of the writer. One of society's many double standards is that, in general, men who choose to avoid becoming a father aren't treated with the level of disrespect often shown to women who choose to avoid becoming a mother.

This article is written from a female point of view, and obviously my commentary isn't.

The COVID-19 crisis has revealed the brokenness of America’s institutions: police violence and the inhumane criminal-justice system, a medical system that lacks infrastructure and essential equipment, precarious employment for an in-debt population getting by month to month, the toxic effects of globalization and climate change. Add to that list middle-class parenting, long an aspirational experience, whose social protections are now showing themselves to be a bit of a charade.

Most of which I've cited in the past as reasons why I didn't want to be responsible for bringing a child into society. Not the only reasons, but reasons. It didn't take a pandemic for me to see the writing on the wall.

While the parents in my life have been openly acknowledging the challenges of parenting during the pandemic, my child-free friends have for the first time been sharing that they are relieved they don’t have children.

Again, a very different experience. I hear some variant of "I'm so glad I never had kids" fairly often. Incidentally, I appreciate the author's use of the adjective "child-free;" many would use "childless," which implies a lack or an emptiness.

Hm... I feel like I should note that I'm not trying to rag on anyone's choices here. I understand that being a parent is very fulfilling for some, and I respect their choices. I would only ask that they do the same for the child-free.

An essay series in The Guardian, called “Childfree ,” explores that decision, with reasoning that runs the gamut: not enough money, focusing on your own life, the climate crisis, being fine with being alone.

Well. The Guardian is one of my usual sources, but I've missed those essays until now. I made it a hyperlink here mainly so I can be reminded to go look at it later. I can't be arsed right now.

In response to a harmless tweet from a parent about how “non-parents have no idea how hard it’s been” to parent during the pandemic, thousands of people chimed in with some version of: Yes, we do—that’s why we don’t have kids.

I know what a terror I was growing up, and I wouldn't subject anyone to that -- least of all ME.

This is hardly the first moment that the idea of marriage and a baby as the primary path for women has come under scrutiny.

Again, the female point of view. Which I'm by no means trying to downplay, but until we work out issues surrounding cloning, it takes two, directly or indirectly, to procreate. I feel the need to add that my own point of view comes from someone who has never had children (please keep "...that you know of" jokes to yourself; they're tiresome, false, and sexist), not like some guys I know who are childfree by way of abrogating their responsibilities. I don't think much of those guys.

For heterosexual parents, the bulk of the child care falls on the mother. The global health crisis has worsened this sexist division of labor, and the long-term effects could damage women’s careers and, despite the best intentions, become a new norm.

Which is another societal double standard that really should be addressed.

For people who were planning to have a child, those plans might now be on hold; the process of seeking fertility treatments, for example, has gotten more complicated as access to medical procedures for elective reasons has been limited.

In a perfect world, as I see it, everyone who wants to have children would, while everyone who doesn't want to wouldn't. As this is not a perfect world, you have people desperately wanting to reproduce who can't, and people who desperately don't getting stuck. Also if I were inclined to pass judgment on anyone, it wouldn't be people who choose to have kids or people who choose not to, but the ones who blow all their resources on fertility treatments when they could adopt.

Taking away a lot of the stigma around adoption might serve to alleviate this disparity. Maybe.

Childless people have long been chastised for being selfish or for not fulfilling a role their body seemingly bound them to.

Here the author reverts to the other word. And selfish? What's more selfish than insisting on bringing children into a deteriorating world? Stipulating that everyone does everything for selfish reasons, if you want to have kids and be less "selfish," then adopt. Also, biology isn't destiny. My body also evolved for chasing prey across the savanna, but I can't be arsed to do that, either.

As a child-free woman in my 40s, I’ve been tasked with taking care of my parents.

Because of course one of the main reasons to have kids is to give you free convalescent care later in life.

One legacy of the pandemic may be less judgment of the child-free.

I won't be holding my breath.

Anyway, it's a point of view, and I thought it was interesting... but again, I admit to some confirmation bias here.

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