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').

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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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

December 22, 2023 at 10:48am

#1061303

Working to Live

Well, it's the start of a new year (by my preferred reckoning); the accursed daystar is happily glaring at me once again, and most of my cold symptoms have lessened. So let's get back to some semi-random articles. This one's from The Conversation, dated last St. Patrick's Day:

Seven tips for finding happiness at work

By now, you should know that I give the side-eye to anything to do with "happiness." But let's take a look at what the author has to say. Be warned: British spelling.

Work, it’s something most of us do though it isn’t always enjoyable.

Not a great start, Professor. Not merely blindingly obvious, but with questionable punctuation choices. I'll let that go, though, because editing is clearly dead.

Whether it’s long hours, gruelling tasks or just the repetitive nature of a day-to-day routine, work can sometimes be something we have to do rather than something we want to do.

Well, there's at least part of the problem right there. As soon as you take the attitude that you have to do something, it becomes a chore, and most of us don't find happiness (or fulfillment or contentment or whatever you're going for) from chores. Say you're invited to a wild party, for instance. There will be lots of your favorite substances there, loads of people of your preferred psychosexual orientation, delicious food, loose morals, great music, etc. Would you be as happy attending if you felt obligated, that you had to go to it? I probably wouldn't. No, better to say "I really want to go to this party" rather than "I have to go to this party."

That is, of course, an extreme. Most work-related things are the polar opposite of that in terms of fun. "I have to finish these TPS reports by 3:00," for instance. Still, turning that around to "I want to finish these TPS reports by 3:00" tricks yourself into greater agency, which, if not exactly increasing happiness, at least decreases misery.

I rarely wanted to work. What I wanted was to get paid. To do that required work. So I'd be like "I want to do these calculations because I'll keep getting paid."

Anyway. That's me. Back to the article.

I was the lead scientist in a government project that looked at how our wellbeing and emotional resilience can change over a lifetime.

It probably doesn't matter much, but "government" here refers to the UK government.

As part of this project, the team, with help from think-tank the New Economics Foundation, identified several things that can reduce stress and enhance wellbeing and happiness – all of which can be applied to the workplace.

Also a UK think-tank. Unlike most think-tanks in the US, this one seems to be a lefty pinko liberal one. Which means I'm more inclined to believe its findings... as long as they don't start in with the whole "mindfulness" crap, or whatever.

1. Be active

Exercise and other physical activities won’t make your problems or stress disappear, but they will reduce their emotional intensity and give you mental space to sort out problems – as well as keep you physically fit.

To my vast shock and surprise, and no doubt everyone else's too (I allow myself one day a year for optimism, and that day was yesterday), I find myself agreeing with this bit.

Not that I'm a model at following it, being a sofa spud most of the time. But having gone through periods of activity and relative inactivity in my life, I can say, unscientifically and anecdotally, that the active periods have been better for my mental state.

Walking to and from work is a great way to create separation from the working day. If that’s not possible you could get off the bus a stop early, make your lunchtimes active or maybe find an exercise class to do before you start work for the day.

Except they lose me at squeezing in yet another activity before work. Sure, I don't have to deal with that anymore, but as a night owl, it was all I could do to wake up on time. Making that earlier to do a new thing wasn't going to happen.

2. Connect with people

If you examine most of the happiness scales, relationships with others come near the top of these lists.

Ugh. People.

Okay, I'm not going to simply dismiss this one, but I will point out that connection feels different for different people. Introverts can't always use advice meant for the extroverted majority; nor, often, can the neurodiverse.

It’s also worth getting to know your colleagues.

Not... not always.

3. Learn new skills

Keeping “cognitively active” is critical to your psychological and mental wellbeing and can provide you with new opportunities in terms of your career development.

Can't argue with this one, either. I do have some objection to "new opportunities in terms of your career development," because I think learning shoudn't have to be merely about how you can be more productive for your employer... but that's me, again. You do you.

4. Stay present

This is all about “being in the moment” rather than in the past or looking too far forward.

here we go.

A more mindful approach to life is something you can practice at any time of the day, it’s just about being aware, noticing your surroundings – the sights, sounds, smells. You can do this while you’re walking, in a meeting or making a cup of tea.

I mean, okay, sure, put that way, I can appreciate what they're talking about.

5. Recognise the positives

Staying present also helps you to recognise the positives in your life – allowing you to be a glass half full rather than a glass half empty person.

How about being a "glass that's twice as big as it needs to be" person? No, seriously, though, while it's probably bad to dwell on the downsides of things all the time, I object to the idea of mentally banishing the negative all the time, as well.

Accept there are things at work or in life you can’t change and concentrate on the things you have control over.

But no argument about that part from me.

6. Avoid unhealthy habits

Given what we know about their long-term consequences, using excessive alcohol or coffee consumption or smoking as a coping strategy for work stress is ultimately likely to have a negative impact on your happiness, even if they seem to provide a quick pick-me-up.

Aw. Too bad.

Note that there's no warning about excessive tea consumption.

7. Work smarter, not longer

Prioritise your workload during working hours and you will have more disposable time to do the things you enjoy.

Okay, whatever. My only objection here is the idea of "disposable time." To me, there's no such thing as that, or "free time." There's only time. If you take the attitude, as I did above, that you're only doing what you want to do, anyway, then all the time is yours. If there's something you think you want to do, but never seem to have the time to do it, then maybe you don't really want to do it.

The more you take control of your work life and get the balance you need, the more likely you will be happier at work.

And let's keep in mind that I never got the balance I needed until I retired.

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