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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 15 hours ago
In response to "Extro Special"

I'll also use the spelling extrovert because if I use extravert my spell checker flags it. In college psychology our professor termed his own words, innervert and outervert because he thought they fit better. I, however, thought they sounded like definitions of belly buttons.

1 Like

Kåre เลียม Enga

about 21 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

Dec 26, 2024 at 4:20pm
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

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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