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 7 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!

Nov 14, 2024 at 2:04pm
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.

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

November 9, 2018 at 1:53am

#945199

Scale

Oh no! Here come the asteroids!

https://www.irishmirror.ie/news/world-news/three-huge-asteroids-fly-dangerously-...

This weekend, three enormous asteroids will make a ‘close approach’ to Earth, NASA has warned.

The asteroids - the biggest of which is predicted to measure up to 30 metres across - will whizz past our planet on November 10.

At around 14:03 GMT, an asteroid dubbed 2018 VS1 will pass the Earth.

This asteroid is predicted to measure between 13-28 metres across - suggesting the asteroid could be five times as tall as a giraffe!

Everybody panic! Time to freak out! Also, what if they're actually ALIENS?!

Or maybe a tabloid is just trying to generate ad revenue? ...nah, couldn't be.

Seriously, though, apparently they're now measuring asteroids in units of giraffe. I guess they used up all the Statues of Liberty. That might also explain why they use both miles and kilometers (as well as meters) in the same article - trying to conserve units.

To be fair, a big part of science communication is the attempt to translate mind-bogglingly big or small things into terms we might be able to comprehend. This many Earths could fit side-by-side across Jupiter, and that many Jupiters could fit across the Sun. Kilometers and miles are just fine for measuring most distances on Earth, but astronomers and cosmologists use larger units like AUs, light years, and parsecs (which are a unit of time, not distance, regardless of what Han Solo said; also, Han shot first). People who study tiny things use angstroms or nanometers or whatever.

A giraffe is a nonstandard unit of measurement, but perhaps it helps with visualization. It's probably a better unit for measuring asteroids than a cat or a paramecium, unless the asteroid is cat- or paramecium-sized. Actually, the latter would be classified as space dust, not asteroid or planet.

People like to smugly tell other people just how insignificant we are. "Look at the universe," they'll say, or something to this effect. "We're barely a speck of dust." Well, compared to the observable universe, we're smaller than a speck of space dust. We're even smaller than an atom, relatively speaking, by this comparison. But it turns out that the smallest particles - smaller than actual atoms; on the scale of electrons or quarks - are close to the same order of magnitude smaller than us as we are smaller than the known universe. In other words, we're right in the middle in terms of size. So maybe not so insignificant after all? Especially since we were able to deduce this fact.

Incidentally, going back to the asteroid thing, let's be clear: these may be close encounters from a scientific point of view, but there's zero chance any of these rocks are going to hit the planet. The closest one will miss by roughly the distance to the moon

, or 30 times the diameter of the Earth.

A miss is as good as 240,000 miles, I suppose.

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