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:

Write_Mikey_Write!

about 2 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 4 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.

August 6, 2021 at 12:06am

#1015220

Trust Issues

Y'all want controversy? I got controversy that shouldn't be controversial.

70 years ago Walter Plywaski fought for atheists’ right to become citizens – here’s why his story is worth remembering

There's a bit going on here, and I'll try to be as clear as I can be as tired as I am right now.

Almost 70 years ago, Plywaski fought for the right of atheists to become U.S. citizens – and won.

Why is this even a thing?

As a scholar of religious and political rhetoric, I believe that Plywaski’s fight is worth remembering. Stories like Plywaski’s give an insight into the discrimination atheists in the U.S. face even today and the role that those professing no faith have had in holding society accountable to the goals of religious tolerance and freedom.

I may not have the bona fides to write about this, but I'm thinking the author of the article does.

Polish native Walter Plywaski, born Wladyslaw Plywacki, spent five years in Nazi concentration camps during the Second World War. After being liberated from Dachau, the Bavarian camp in which 41,500 prisoners died, he worked as an interpreter before immigrating to the U.S and serving four years in the U.S. Air Force.

Just leaving this here to point out that the dude served in the USAF.

In August 1952, Plywaski petitioned for U.S. citizenship while in Hawaii. All he had left to do was say his oath of allegiance.

Plywaski, however, was an atheist. He informed the judge that he could not sincerely end the oath with the words “so help me God” and requested an alternative.

Simple enough, right?

Judge J. Frank McLaughlin reportedly asked Plywaski to consider what it says on the back of U.S. coins: “In God We Trust.”

This is where I get a little hazy on things. IGWT was introduced to US coins in 1956, a reaction to McCarthy-era demagoguery. Now, I can believe that it took more than four years for his citizenship application to get through the system even then, but if that's the case, the article could be more clear on that.

McLaughlin then denied Plywaski citizenship, justifying his decision by proclaiming, “Our government is founded on a belief in God,” and accused Plywaski of “seeking admission on your own terms.”

Arrant nonsense. Our government is an outgrowth of the very secular Enlightenment, and it's right there in the First Amendment: "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof..." It's right up front; you couldn't miss it if you tried. Now, I'm not a constitutional scholar, obviously, but I think a plain reading of the text is pretty clear. And you can't have freedom of religion if you don't also have freedom from religion.

I've heard before the phrase: "The US is a Christian country." It most certainly is not. It's a secular country with a Christian majority. Moreover, it's a republic, which means that there's an obligation to protect the rights of the minority. If Christianity were a minority (and we seem to be heading in that direction), then the rights of its adherents must be protected as thoroughly.

McLaughlin, however, stood his ground. He argued that the case was not about religious freedom but about whether Plywaski “believes in all the principles which support free government,” which according to McLaughlin included a belief in God.

Free government is incompatible with theocracy.

In January 1955, Plywaski won his case and became a citizen.

See, that was still before IGWT became mandated on currency. Granted the motto preceded the (unconstitutional) law about it, but still, something's off and I can't be arsed to figure out what right now.

The article goes on to describe other cases of clear religious (or anti-religious, as the case may be) discrimination, even after that precedent was set.

My research shows how the discrimination against atheists fits with what I describe as a deeply ingrained and coercive theistnormative mindset that frames democratic societies and good citizenship as being tied to belief in a higher power.

Utter tosh.

A person's beliefs (or lack thereof) are their own: this is, as I noted above, enshrined in our founding documents. What matters is a person's character -- what they do, how they treat other people, animals, etc. -- not what they profess to believe (or not believe). I'd argue that a lot of atheists can be more compassionate, because they're not being told to be by some ancient text or modern epiphany, and because they direct it where it belongs, at other beings, not for the glory of an invisible sky-wizard.

Of course, some atheists are assholes, just like some religious people are assholes, because we're all still just people.

And despite the glib platitude "there are no atheists in foxholes," plenty of atheists and agnostics have served in the military, putting their lives at risk, and continue to do so -- with absolutely no hope of an afterlife, whether Heaven, glorious Valhalla, or anything else, but just to protect the freedom of their fellow citizens to believe, or not, as their conscience sees fit.

Believe what you will. But belief in a deity is useless if it's coerced.

One-Sentence Movie Review: The Suicide Squad

If, like me, you're a fan of comic books, movies derived from comics, action, and dark humor, this movie is for you; I loved every damn minute of it.

Rating: 5/5

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