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 11 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 13 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.

June 6, 2020 at 12:17am

#985086

Save

I really do not like the website I'm going to link today. But it's the link I've got and I'll do what I can with it.

https://twocents.lifehacker.com/its-time-to-live-on-a-budget-even-if-youre-finan...

It's Time to Live on a Budget, Even if You're Financially Secure

It's always been time to live on a budget. Maybe unless you're Jeff Bezos or Elon Musk. Chances are you're not. But hey, if either of you are reading this: Hi! Jeff, Elon is kicking your ass in the space business. Elon, Jeff is kicking your ass in the everything else business. Both of you get on the ball.

There are two big unknowns in all of our lives right now: What’s going to happen with the COVID-19 pandemic, and what it will mean for the economy.

This article is from a month ago, so there's more than two unknowns.

Some of us are already having to rethink our spending habits to make ends meet. Others may feel more financially secure thanks to high salaries, six-month emergency funds or the ability to work remotely. But when recessions come, they tend to come for everybody—so it’s time for all of us to reevaluate our budgets and, potentially, start saving as much as we can.

I can't find it today, but at some point last month, when 30DBC was going on, I saw an article that basically berated all of us for not spending money. I wish I'd remembered to save it, but I didn't. From what I recall, it was some corporate media site like CNN or something, trying to shame us as hoarders if we dared to hang onto cash instead of feeding the capitalist machine. I probably would have read more of that article if I hadn't gotten a pop-up requiring me to subscribe. I laughed and closed the tab.

It is true that our economy depends on the flow of cash. It is debatable whether what we had in the Before Times, where many of us had crushing debt and yet continued to spend more, more, and more, all while slaving away at a job that was able to pay a relative pittance because if you got fed up, there was no guarantee that you could even get another job, let alone one that paid more. Even with what was, at the time, remarkably low unemployment numbers, employers still held all the cards.

Now they hold all the cards and have a few aces tucked in their boots as well.

Deficit spending may work for governments. Or it may not, in the long run; I don't know because macroeconomics is really above my pay grade. Hell, the "experts" in the field often disagree. I mean, climate scientists are mostly all in agreement that we're boned, but economists? Ask three economists what needs to be done to fix the economy and you'll get six different answers. But I'm almost certain that deficit spending is the road to ruin for individuals and families. It's a hamster wheel, moving fast but without many people actually getting anywhere.

If you're in debt of any kind, if you have expenses you can't cut, if you rely on your employer for health insurance... they OWN you.

So, no, I strongly recommend that you don't listen to the alarmists going "but if people don't spend, the economy will collapse!" I mean, yeah, maybe it will, but then what? There you are in a shitty economy with no savings because you spent all your money on scented candles from Amazon.

So what does Lifehacker recommend? With the caveat that I wouldn't blindly do anything that they recommend, you could always read the article. Here's a summary, though:

Treat your income like it’s temporary

I'm not sure if that's helpful or not. I do believe that it's a strong reason to save: if you think of your income as temporary, then you'd better have something squirreled away for when you don't have income.

Saving money now is worth more than spending money later

I think it's a little more complicated than that. I'd say that saving money now is more important than buying that shit you found that normally sells at a 1000% markup but is on sale for only an 800% markup. Remember: If you spend $80 on a $100 knickknack, they'll have you believe that you "saved" $20, when in reality you spent $80.

Try budgeting for 30 days, then reevaluate

What? No. Budgeting is an ongoing process. Spend the first 30 days just tracking your expenditures. Find ways you can cut back. Lather. Rinse. Repeat.

Look, I'll be the first to admit that I spend more than I should, and like most people, I've been known to make impulse purchases. Having money in the budget to do that, if you can, is important to one's mental health.

Dickens probably said it best, at a time when money bought a lot more and long before the British finally decimalized their currency. But the idea remains valid:

Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness.
Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

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