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:

Turkey DrumStik

about an hour ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

a day 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

a day 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

February 18, 2019 at 12:11am

#952326

Welp... We're Boned

At some point, birthdays stop becoming about growing up and start becoming about growing old. Personally, I think that point is age 21 in the US, because drinking.

But now I find they're less about growing old and more about being one year closer to death.

Hence, today's article:

https://www.bbc.com/news/world-47030233

What are the biggest threats to humanity?

Oddly, social media didn't even make the list. Neither did commercials, vegans, or SUV owners.

Now I'm picturing a vegan ad agency executive driving her SUV while instagramming. Don't tell me that never happens.

Volcanic threats

But while the prospect of a super-volcanic eruption is terrifying, we should not worry too much. Super-volcanoes and other natural disasters, such as an asteroid striking Earth or a star exploding in our cosmic neighbourhood, are no more likely in 2019 than any other year. And that is not very likely.

Before the 2017 eclipse, I noted that its track passed over or very close to the Yellowstone Caldera. So I tried to start a viral rumor on the internet that the massive combined gravitational force would trigger the Yellowstone Supervolcano. (Just to be clear, this was, is, and will continue to be bullshit.) It didn't take. I was devastated. The world was not.

Growing threats

The threats are complex and diverse, from killer heatwaves and rising sea levels to widespread famines and migration on a truly immense scale.

And this is why I still celebrate birthdays. Well, apart from the existence of tequila. Because every birthday means 1 year less chance of me being alive when shit goes south.

Also increasing are the potential risks from novel technologies such as artificial intelligence (AI).

Remember yesterday's entry about Lone Asshole Theory? This particular doom is related to that. Sure, we can try to put limits on AI - but all it would take would be one lone asshole ignoring those limits.

Of course, I think the "threat" from AI is way overblown. Fuckers can't even get autocorrect to work right.

Another threat is the possibility of a nuclear war.

The 80s called. They want their fearmongering back.

Increasingly dense and mobile human populations have the potential to see new influenza strains spread easily. And this raises concerns about a future outbreak like the 1918 Spanish Flu, which killed up to 50 million people.

However, widespread vaccination programmes and other disease prevention measures help reduce this risk.

I think my next attempt at viral scaremongering should be how the anti-vax "movement" was started by the Zero Population Growth (ZPG) people (or possibly the Illuminati or the Freemasons) in an attempt to reduce world population.

With almost eight billion people living on Earth, we are increasingly reliant upon global systems to sustain us. These range from the environment that provides us with food, water, clean air and energy, to the global economy that turns these into goods and services.

Maybe it's not a conspiracy theory. Study it out, sheeple.

There is an old story of King Canute of the Danes commanding the sea to retreat. He knew he would be unable to hold back the tide and a similar sense of powerlessness can easily overtake us when we consider potential future catastrophes.

However, the truth is that the Danes have been pushing back their shoreline for generations: building dykes and draining marshes to protect themselves from the oncoming tide.

Ah... the power of civil engineering. Unfortunately, I'm retired.

4 Like • Comment

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