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 16 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 18 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 17, 2020 at 12:01am

#978319

The Pursuit of Grumpiness

In these days of plague, economic disaster, and impending environmental collapse, it's important to remember to maintain a negative attitude.

No, really.

https://www.bbc.com/future/article/20160809-why-it-pays-to-be-grumpy-and-bad-tem...

Why it pays to be grumpy and bad-tempered
Being bad-tempered and pessimistic helps you to earn more, live longer and enjoy a healthier marriage. It’s almost enough to put a smile on the dourest of faces.

Oh, believe me, this makes me happy. See? This. This is my happy face. (By way of illustration, I look something like this

right now.)

The pressure to be positive has never been greater. Cultural forces have whipped up a frenzied pursuit of happiness, spawning billion-dollar book sales, a cottage industry in self-help and plastering inspirational quotes all over the internet.

All of which have the opposite effect than intended on me.

Now you can hire a happiness expert, undertake training in ‘mindfulness’, or seek inner satisfaction via an app. The US army currently trains its soldiers – over a million people – in positive psychology and optimism is taught in UK schools. Meanwhile the ‘happiness index’ has become an indicator of national wellbeing to rival GDP.

Hire... a happiness expert? No wonder society needed a reset.

The truth is, pondering the worst has some clear advantages. Cranks may be superior negotiators, more discerning decision-makers and cut their risk of having a heart attack. Cynics can expect more stable marriages, higher earnings and longer lives – though, of course, they’ll anticipate the opposite.

As I've always been saying: Expect the worst, and then you can only be pleasantly surprised. Either you're right, which makes you feel good; or you're wrong, which also makes you feel good because the worst didn't happen after all.

Good moods on the other hand come with substantial risks – sapping your drive, dimming attention to detail and making you simultaneously gullible and selfish. Positivity is also known to encourage binge drinking, overeating and unsafe sex.

Now, see? If I were a jovial, positive, upbeat person, I'd binge drink and overeat even more than I do now! On the other hand, maybe I'd have sex.

The article goes on to describe the actual science behind this, which I'm going to choose not to be skeptical about (even though it takes a detour into evolutionary psychology speculation) because dammit, I want affirmation every once in a while. But, you know, feel free to grump about it yourselves.

All these physiological changes are extremely helpful – as long as you get a chance to vent your anger by wrestling a lion or screaming at co-workers. Sure, you might alienate a few people, but afterwards your blood pressure should go back to normal. Avoiding grumpiness has more serious consequences.

Alienating a few people? Sounds like heaven.

In fact, scientists are increasingly recognising that grumpiness may be beneficial to the full range of social skills – improving language skills, memory and making us more persuasive.

See? If I weren't grumpy, I'd have no social skills at all.

Then there's more science describing some more studies to explain just how happiness is a Bad Thing.

So the next time someone tells you to “cheer up” – why not tell them how you’re improving your sense of fairness, reducing unemployment and saving the world economy? You’ll be having the last laugh – even if it is a world-weary, cynical snort.

Now sod off. I'm social-distancing.

3 Like • Comment • View 2 comments

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