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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Turkey DrumStik

about 3 hours 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

Nov 14, 2024 at 12:10pm
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.

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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

January 21, 2019 at 12:31am

#950101

We Need Space

Today in the US is Martin Luther King Day, a federal holiday celebrating the Reverend Martin Luther King Jr., a civil rights activist best known for advancing civil rights through nonviolence and civil disobedience. In August of 1953, King delivered his famous speech “I Have a Dream” calling for civil and economic equality for all Americans and the end of racism. On this Motivational Monday, write a speech advocating for something you’re passionate about and use the phrase “I have a dream” in your entry.

We need space.

We need space, not because going there will solve all of our problems, but because it is there.

We need space, because we're only insignificant if we allow ourselves to be.

Fifty years ago this year, humans walked on the Moon for the first time. As I write this, that Moon is eclipsed. Our satellite slipped into our shadow and darkened.

During a lunar eclipse, the Moon glows red. That red is sunlight, scattered by the atmosphere. If we could stand on the Moon right now, the Earth would be a huge red ring in the dark sky, the light of every sunrise, every sunset, every twilight illuminating the lunar orb.

Some say we shouldn't bother with space, that we should solve our problems here, first. Well, we've had fifty years to solve our problems, and we still have problems.

We shouldn't go to space, they say, because there's still poverty, still hunger, still war, still crime.

But do we stop searching for a cure for pneumonia just because we haven't found a way to end cancer? Do we stop convicting rapists because there are murderers out there? Do we stop feeding ourselves because someone else is still hungry?

There are over seven billion of us. I think we can work on more than one thing at a time.

Do you think it's a waste of money? Why? Are you under the impression that we stuff dollar bills into rockets and blast them into orbit, never to be seen again? No. That money enriches our economy. It helps all of us, rich and poor alike.

The things we've already learned have improved our quality of life immeasurably. And there's still much more to learn. Everything we learn helps us to prosper. Everything we find advances science.

Is it difficult? Yes. Is it dangerous? Oh, yes. But doing nothing, staying here, losing our curiosity and our drive to explore - that's not only dangerous, it's suicidal.

Maybe you think we don't deserve to survive. Maybe, because we've produced serial killers and pedophiles, and despoilers of the environment, and hedge fund managers, you would condemn the whole human race to purgatory. Well, see, by even pronouncing that opinion, you're proving the opposite: because you are not a rapist or a thief or a drug pusher, and you condemn those activities, you're showing that we do, in fact, deserve to survive.

I have a dream that, one day, someone will stand on the Moon, look up, and witness the Sun slipping behind the limb of the Earth. That we'll have a thriving colony on Mars. That, eventually, some of us will leave our home forever, sailing out to explore the billions of stars in the galaxy, and perhaps even the billions of galaxies beyond.

Will we find other life? Almost certainly. Will we find Vulcans or Klingons? Almost certainly not. Wouldn't it be better to know than to speculate?

This is what we're for. Let's do it.

Give us space.

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