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

April 8, 2023 at 10:40am

#1047803

Uncivil

I do like the occasional "well, actually..." piece, so here's one from Mental Floss:

7 Misconceptions About the Civil War

By "Civil War" they mean the American one in the 19th century, not the one we're in now, not another country's war, and not the Captain America sequel.

I should note that I didn't fact-check this, but I did go to some of the links and it looks like they did their due diligence. Just don't blame me for any continued inaccuracies on their part.

The American Civil War is a pivotal and ugly moment in American history, but it's more misunderstood than you would think.

I wouldn't think. Given the near-deific significance it's accorded by Americans, northern and southern alike, I'd expect people to get its history right. But as is often the case, I overestimate people—or, at least, ACW fanatics.

1. Misconception: Lincoln’s policies enjoyed widespread support in the North.

I find that, in hindsight, an event in history is usually more clear-cut than it was at the time. I wouldn't expect that everyone in the North supported Lincoln, or that everyone in the South wanted to secede. It's kind of like "the colonies wanted to break away from oppressive England" without considering how many colonists wanted to keep saying "pip-pip-toodleoo."

Take a look at the article for some well-sourced examples.

2. Misconception: Robert E. Lee and Jefferson Davis were staunch secessionists.

I've known this was wrong for a long time (again, details in the link), but pointing it out now seems to be a failure to read the room.

3. Misconception: The Emancipation Proclamation ended slavery

When President Abraham Lincoln delivered the Emancipation Proclamation on January 1, 1863, it declared: “[All] persons held as slaves within any State or designated part of a State, the people whereof shall then be in rebellion against the United States, shall be then, thenceforward, and forever free.”

Well, clearly not, as slavery still exists. Even after the 13th Amendment (which the article covers).

4. Misconception: All amputations were done without anesthesia.

Sure, but it makes for a more compelling movie to ignore that.

5. Misconception: Only men fought during the Civil War

Pretty sure that if you look at wars throughout history, there were almost always women finding a way to join in the fun.

6. Misconception: Abraham Lincoln was the keynote speaker on the day of the Gettysburg Address.

Honestly, this one's pretty interesting. Not surprising to me, but has details I wasn't aware of.

One thing that you might not know about the address is that Lincoln wasn’t pegged to be the main speaker on that day. That honor belonged to Edward Everett, a distinguished scholar and orator who took the stage before the president.

Everett’s speech would go on for around two hours, totaling upwards of 13,000 words...

After Everett finished his speech, the president shook his hand and told him, “I am more than gratified, I am grateful to you.” Then the Thunder-Stealer-in-Chief rang out with “Four score and seven years ago ...” and made Everett’s magnum opus a historical footnote in under 180 seconds.

The original Rap Battle.

7. Misconception: The war was fought entirely in the U.S.

Well, duh, because technically, the CSA wasn't part of the US; hence the war.

But no, this part's mostly about naval battles.

I'd also add misconception #8: Everyone knows the war is over.

4 Like • Comment • View 4 comments

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