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 9 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 11 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.

February 1, 2024 at 11:19am

#1063273

And It Was All Yellow

Well, here we are: the worst month of the year. At least it's shorter than other months. Wait, this is a leap year? Rats.

No sense dwelling on it. I'll just pick another site from my queue to blog about, and move on.

One perfectly legitimate reason to avoid having kids, besides the obvious ones, is that, at some stages, they ask awkward questions that may be hard to answer in an age-appropriate manner.

Like this one.

Why is Gold yellow? Spoiler alert: Einstein again
It turns out you don't have to look further than gold jewelry to experience Einstein's relativity.

I don't think I ever asked my dad that particular question, but I did ask the classic "why is the sky blue" one, and his lengthy scientific explanation went right over Kid Me's little head. I think I might even have fallen asleep.

Eventually, I understood the reason for the effect, at least to some degree. And I've often wondered why gold appears, well, gold-colored, when most other metals (with some obvious exceptions, like copper) are various shades of gray. Just haven't wondered enough to be arsed to look it up. Then this article somehow got my attention.

You’ve probably not given it much thought, but the reason why gold is yellow (or rather, golden) is deeply ingrained in its atomic structure — and it’s because of something called relativistic quantum chemistry.

Now, here's the thing: Generally (pun intended), relativity deals with large-scale things, while quantum mechanics deals mostly with the smallest. I'm told the two theories are both well-supported; that is, predictions they make turn out to match observed evidence. And yet, they're incompatible with each other (again, just going by what I've heard, here). So for me, the phrase "relativistic quantum chemistry" is itself a shiny gold thing that I just have to pick up.

Simply put, because it’s a very large atom, gold’s electrons move so fast that they exhibit relativistic contraction, shifting the wavelength of light absorbed to blue and reflecting the opposite color: golden yellow.

And there you have it: the answer. Refreshing to see that at the beginning of an article. Of course, it goes on to explain that shit further, which is also good.

Incidentally, this is an entirely different phenomenon from why the sky is blue, but I can't help but notice the parallels: blue light from the sun gets preferentially scattered by air molecules, turning the sky blue and making the sun appear yellower.

Read on for a more in-depth look into the fascinating chemistry that gives this symbol of wealth and luxury its prized color.

At this point, it's less a symbol of wealth and luxury than one of overindulgence and greed. Still pretty, but come the apocalypse, all that gold you've been hoarding will be mostly useless. It's not like any survivors will need it for its amazing electrical conductivity.

Anyway. I won't bore you with too much from the rest of the article. I do think it's very accessible—probably too much for a toddler, but not for a teen or a grown-ass adult—so give it a look if, like me, you want to finally get that stubborn curiosity looked at.

5 Like • Comment • View 4 comments

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