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 10 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 12 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 23, 2022 at 12:01am

#1029364

Unbound

I don't have a horse in the race on this one, but I'm sharing it anyway in the interest of reducing nonfictional drama in the world.

Family estrangement: Why adults are cutting off their parents
Polarised politics and a growing awareness of how difficult relationships can impact our mental health are fuelling family estrangement, say psychologists.

You'll pardon the British spelling, as the article is from the BBC.

It was a heated Skype conversation about race relations that led Scott to cut off all contact with his parents in 2019. His mother was angry he’d supported a civil rights activist on social media, he says; she said “a lot of really awful racist things”, while his seven-year-old son was in earshot.

Right away I can see someone arguing, "Isn't that just like cutting off your kid if you find out he's gay?" No. No, it is not. Gay isn't a choice. Being a racist is most definitely a choice. Besides, you have an obligation to your kids to at least raise them to adulthood. One could then argue that we also have obligations to our parents. I disagree, but I can see both sides of that argument. In my opinion, you have no obligation to be around anyone whose values are trash, family or otherwise.

Despite a lack of hard data, there is a growing perception among therapists, psychologists and sociologists that this kind of intentional parent-child ‘break-up’ is on the rise in western countries.

Good.

The survey showed more than one in four Americans reported being estranged from another relative. Similar research for British estrangement charity Stand Alone suggests the phenomenon affects one in five families in the UK, while academic researchers and therapists in Australia and Canada also say they’re witnessing a “silent epidemic” of family break-ups.

"Epidemic" is a loaded word, especially now. It implies that there's something wrong with the practice. While individual circumstances vary, sometimes you have to withdraw for your own mental health.

On social media, there’s been a boom in online support groups for adult children who’ve chosen to be estranged, including one Scott is involved in, which has thousands of members.

You have the freedom to choose your own family. I know this goes against the grain of long-standing tradition and maybe even some evolutionary drive, but I stand by this assertion.

One of the most common reasons for this is past or present abuse by the parent, whether emotional, verbal, physical or sexual.

And that's absolutely a valid reason.

Both experts believe at least part of the context for this is increased political and cultural polarisation in recent years. In the US, an Ipsos poll reported a rise in family rifts after the 2016 election, while research by academics at Stanford University in 2012 suggested a larger proportion of parents could be unhappy if their children married someone who supported a rival political party, which was far less true a decade earlier.

This is my shocked face.

“Not needing a family member for support or because you plan to inherit the family farm means that who we choose to spend time with is based more on our identities and aspirations for growth than survival or necessity,” he explains. “Today, nothing ties an adult child to a parent beyond that adult child’s desire to have that relationship.”

One of the questions often asked of the childfree is: "But who will take care of you when you're old?" Listen, bringing another life into the world, this world, the way it is now, is ethically questionable at best. If you're only (or mostly) doing it so someone will change your Depends when you're 90, well, I can't think of anything more self-centered.

And then they have the chutzpah to call us "selfish."

Of course, all of this also has an impact on the parents who have, often unwillingly, been cut out of their children’s – and potentially grandchildren’s – lives. “Most parents are made miserable by it,” says Coleman. As well as losing their own footing in the traditional family unit, they typically “describe profound feelings of loss, shame and regret”.

If you're racist, if you're abusive, if you support racist and abusive policies... you should feel bad.

But he’s unlikely to reconcile with his own parents, unless they recognise they’ve been racist. “The whole ‘blood is thicker than water’ - I mean, that's great if you have a cool family, but if you're saddled with toxic people, it's just not doable.”

There has been some argument about what "blood is thicker than water" really means (though not usually the kind of argument that results in cutting people out of your life). Usually, it's taken to mean that genetic family is more important than any other kind of relationship. If that's the interpretation, the saying never sat right with me, perhaps because I was an adopted child. My relationship with my parents -- the people who raised me -- was mostly fine, though with the same kinds of friction you see in any family. I don't give a shit about trying to find biological relatives, and all the movies and TV shows where people have this terrible urge to do that always piss me off. It always leads to more drama, which is great in a story, and a massive pain in the ass in real life.

Like I said, family is who you choose. You'll still have arguments, but as almost any parents of an adopted child will tell you, we make our own bonds.

3 Like • Comment • View 2 comments

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