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 14 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 16 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.

April 8, 2020 at 12:02am

#980499

Awkward

Well, this is the last David Wong article from Cracked in my queue, and one I found interesting as someone who has trouble navigating social situations that don't involve buying beer. As usual, he has insights I hadn't thought of.

https://www.cracked.com/blog/15-things-socially-awkward-people-need-to-know/

15 Things Socially Awkward People Need To Know

I don't know if I'm "socially awkward" because I don't get out much to find out.

1 You Don't Know Yourself Without Constant Feedback From Others

It's impossible to know what impression you're making on others without gathering a bunch of data -- seeing how people react, using them as your mirror.

But to do that, I'd have to be around - ugh - people.

2. The Rules Of Human Interaction Make No Sense, And You Can't Just Intuit Them

It doesn't matter how smart you are. Unless you get tons of practice being around people, you'll be shit at it. The reason for this is both obvious and largely unacknowledged: The rules of human interaction are nonsense by design, to give a leg up to people who spend a lot of time socializing.

Me: The only way to win is not to play.

3. Social Anxiety Can Be (Somewhat) Healed By Socializing

You have to understand that other people aren't grading your ability to avoid embarrassment. They're grading you on how well you react to embarrassing situations. But that is a learned skill, and you just have to get practice.

So much makes sense now.

4. Pop Culture Is A Horrible Teacher

This one hurts to even think about, because I did it myself and inflicted untold damage as a result. Sitcoms granted me a great ability to snap back at people in a way that made bystanders laugh and the target hate me forever. Yeah, it turns out that a glorious, sarcastic smackdown will win an argument in a TV show, but never, ever win one in real life.

That's okay. I'm much better at snappy rejoinders online, where I can think about them before burning someone with them.

5. Simply Avoiding People Isn't An Option

If you are the rebellious type and decide you'll just sit this game out, well, that's up to you. But being bad at talking to people will cost you jobs, relationships, and life opportunities from now until the grave.

That's a chance I'm willing to take. Not that I have any choice.

6. If You're Never Around, People Will Invent A Fake Version Of You In Their Minds

... And you probably won't like what they come up with.

Another argument for avoiding people.

7. You Will Be Assigned A Social Role, Whether You Like It Or Not

You, therefore, might get the role of "nerdy punching bag" or "arrogant jerk" or "luckless loser" purely because they need that in their story.

Pretty sure I'm already a negative example.

8. Your Abuse Is Their Bonding Ritual

They torment you as a group activity to make themselves feel closer to each other. You're just the prop in their ceremony.

Again, who needs 'em?

9. Everyone Is Playing A Character At All Times

The world in general will never make sense until you truly understand this: Everyone's actions are the result, not of what they want to do, but of what persona they are trying to project to the world. I would emphasize the word "trying" there. Their decisions are ultimately the result of a bitter conflict between:

A) The person they are;

B) The person they want to be;

C) The person they think society expects them to be.

This kind of harks back to that "authenticity" entry from a while ago.

10. Everything Is A Subtle Game Of Intricate Lies

So these aren't lies in the sense that they're just stating incorrect information, but they are obscuring your true feelings and intentions to make interactions more comfortable for the other person. Once again, this convoluted dance is a learned skill that requires thousands of repetitions to perfect. But just to be clear: if you don't know exactly when and how to lie, you don't know how to function as a human being.

It's always bugged me that people can't be straightforward about anything, but I'm not always straightforward either, so this has the ring of truth to it as well.

11. The Ability To Spot And Call Out These Lies Is An Empty, Pointless Skill

I spent years expecting everyday interactions to operate according to the rules of my robot logic, and thought I was a genius for pointing out when they didn't. Instead, it just exposed my inability to pick up subtle social cues and made me unpleasant to be around. At that point, the best-case scenario is that you parlay it into a career in comedy.

And that's why the internet is an unmitigated boon to people like me. I can do comedy and never have to see the audience or have them throw rotten fruit at me.

12. If You Think Someone Is Perfectly Good Or Awful, You Simply Don't Know Them

You know that thing I said earlier, about how people will just invent a character for you if you're not actually there? Well you're doing the same right back to them. The overwhelming majority of what you think you know about other humans is, in fact, just details your imagination filled in upon glimpsing their shadow.

In spite of, and in direct contradiction to, what I said above, I don't dislike people. But I also know that no one is purely good or evil; we all have our high and low points. I'm okay with this, too. Which leads us to...

13. "People Are Trash" Is An Immature, Naive Worldview

In my case, the biggest barrier was getting so disappointed in people when they didn't live up to my expectations that I had a "one strike and you're out" standard. Every failure was a bitter, unforgivable betrayal. It turns out that knowing how to react when real people disappoint you also takes practice.

I actually think most people try to do good, at least to those they care about. Some people are simply more caring than others.

14. If Your Relationship Is 100% Free Of Friction/Conflict, It Just Means You're Not Close

What we discovered in the era of social media is that loose/distant friendships are more comfortable than tight ones, in the same way that a tent is easier to construct than a house. You may not even see the problem at first -- it's got walls and a roof, right? -- but you'll see it when the storm comes.

Things would be so much simpler if people would just do what I want them to.

15. No One Is Perfect At This

The biggest mistake the socially awkward make is in the assumption that everyone else finds it easy, that their every offhand word or joke represents perfect intention and strategy. Being good at this means understanding that very few people are great at it, and most people are just muddling through. Sometimes there is no more powerful people skill than the ability to just let shit slide.

I've gone through periods where I was convinced that everyone else had their shit figured out and they were just wondering why I didn't. I learned a long time ago that this attitude is wrong and self-absorbed. Also, I have a lot of shit figured out. I'm still going to go out of my way to avoid interpersonal drama.

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