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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

August 21, 2024 at 10:32am

#1075545

Les Lois Bizarres

I figured it'd be important to read this Cracked article, because I'd like to avoid running afoul of the gendarmerie when I go there.

5 Normal Family Activities That Are Illegal in France
Want to take a paternity test? Nope. Straight to jail

France’s official motto is “Liberté, égalité, fraternité.” They’re a little iffy on the liberté part of that, at least by the standards of places who really care about freedom.

Um, have you seen the US?

For example, France bans public school students from wearing religious symbols.

From what I've heard, there were good reasons for the ban.

In the U.S., it would be unthinkable to have any national law on what schoolkids can wear.

Um, not the same thing. France is comparable in size to one of our larger states. Smaller than Texas but bigger than California. Texas and California have state guidelines for schools. That particular ban isn't in place there, but we have other quirks.

But several strange French laws concern families and kids...

I'm absolutely certain that other countries find some of our laws just as strange.

5. Paternity Tests

In France, a court may order a paternity test as part of a legal proceeding, but short of that, they forbid citizens from testing paternity.

Not my problem and, again, we have weirder laws.

4. Naming Your Kid What You Like

In France, courts can rule that parents named their child wrong and order the name changed. In 2015, for example, a court learned that parents were naming a child “Nutella,” and the court renamed the girl “Ella,” to spare her from derision.

Yeah, no, I'm gonna side with France, here.

3. Changing Your Surname

In France, however, there is no simple process for changing your last name. You can petition a court to change your name, if your motive matches one of the few acceptable ones, but you can’t just fill a form and change your name because you feel like it.

This just in: different countries have different laws and customs. Who'd have thought?

2. Disinheriting Your Children

In most places, if you die without a will, your money goes to your next of kin, but you can instead write a will leaving your money wherever you like. Not in France, where you must leave money to your children, no exceptions.

Eh, I don't see this as much different from estate taxes. Either way, there's a law concerning the disposition of a portion of your estate; the only difference is who that has to go to.

But I can see this becoming especially enraging for some men if they spent the kids' entire lives not knowing if they were genetically related, as per #5 above.

1. Swimming Trunks

You may have heard tales of French nude beaches, where every part of your body can go proudly on display. From these, you might think France is a lot more free than other countries when it comes to what you can wear when swimming.

It's not just France. I think the US is one of the outliers here, also known as "the strange one."

When you go to a French pool, you are not allowed to wear swimming trunks. You instead must wear swimming briefs, or what we’d call a Speedo, giving everyone a proper look at the contours of your bulge.

I started adding "Speedos" to the shopping list for my upcoming trip, then crossed it out and wrote NO SWIMMING. You're welcome.

On the other hand, if you want to show off much more of yourself and wear a thong, nope, pools ban that as well.

In summary, lots of rules that make sense in one culture seem strange to others. Part of the reason to travel is to experience these differences. But it's a good thing to have fair warning beforehand, so you don't, say, bring chewing gum to Singapore and end up getting caned.

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