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 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

a day ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

December 2, 2024 at 9:04am

#1080734

The Sans of Time

Today, we have an article from Fast Company about something of great worldwide import.

How Comic Sans became the Crocs of fonts
After 30 years of abuse, Comic Sans is ready for its redemption.

Objection! Comic Sans never deserved the opprobrium heaped upon it by self-proclaimed font snobs, whereas Crocs deserve every criticism and then more on top of that.

Comic Sans has turned 30, and it’s done being your punch line.

I have long said that it should be the Official Sarcasm Font of the Internet.

For three whole decades, Comic Sans cowered at your reproaches and winced at your jokes.

That's obviously poetic license, but I have occasionally wondered how the anticomicsans vitriol might have affected the poor, innocent font creator.

It barely flinched when Google’s practical joke made sure that searching for “Helvetica” would render all results in Comic Sans.

Okay, I hadn't heard of that, but that's legitimately hilarious.

But Comic Sans has just hit the big 3-0—and it’s ready for its second act.

Great, make the rest of us feel even older than we are.

Don’t take it from us. Take it from various studies that have been done on the subject of “turning 30.” And from the three experts who contributed to this story and said that turning 30 marks a period of introspection and change.

And this is where it goes from whimsical poetic license to stretching a metaphor beyond its elastic limit. But, whatever, I'm entertained.

For starters, Comic Sans wants you to know it wasn’t ever meant to be taken seriously. Vincent Connare, who was then a typographic engineer working at Microsoft, created the typeface in 1994.

Well, that answers part of my musing above, musing that never reached the level of "why don't I just google it?"

Most of us were blissfully offline back then, so Microsoft had devised a program called Microsoft Bob to teach people how to use computers.

I'm pretty sure 1994 was the year I first obtained an internet provider. But I'd been using computers for at least 15 years before that, both for work and recreation.

Well, "recreation" included learning how to code, which, let me tell you, was a lot harder to do before the internet. We had to buy books.

It also included playing early video games, which is why I'm not good at coding to this day.

Comic Sans was inspired by the comic books Connare had lying around in his office.

Hence the name. I always figured it was from comic books, not funny-ha-ha comics.

Comic Sans appeared on restaurant menus, funeral announcements, official government letters, bumper stickers, business signs—so many places, in fact, that there’s a popular subreddit on the topic. Even the Vatican used it in 2013 to commemorate Pope Benedict XVI.

Like I said, I don't hate the font, but if I saw it everywhere, I'd learn to. Like with Crocs.

The boom continued well into the early aughts...

Noughties, dammit!

...at which point Microsoft released a licensed version of Comic Sans in 2010.

And the hate might also have spilled over from a generalized dislike of Microsoft, who have definitely made some... questionable... design decisions. Like Clippy.

Over the past few years, the font has become a favorite among people with dyslexia because “the letters can appear less crowded” and “it uses few repeated shapes, creating distinct letters.”

I'd heard that the font was originally designed to be dyslexic-friendly. Perhaps that was fake news. Designed that way or not, it seems to be so. Perhaps some people got less vocal about their distaste for CS for fear of being labeled ableist.

Like the Eiffel Tower (which drew a slew of protests while it was still in construction), or that Mariah Carey Christmas album, the typeface has become nothing short of iconic. (Though we can agree to disagree on the Mariah front.)

Oh, yeah, damn right we disagree. I have no hate for Mariah Carey and acknowledge that she's talented, but that album is about 70% of the reason I don't go out in public in December.

Today, Comic Sans is the Crocs of fonts. First we hated it, then we loved to hate it, now we kinda, maybe love it because we’re experiencing it through a different lens.

"We" my ass.

My issues with the writing, and the author's questionable taste in music and footwear, aside, I'm glad to see Comic Sans finally getting some... well, not love, exactly, but less hate.

I still say it should become the Sarcasm Font, but no one listens to me.

5 Like • Comment • View 1 comment

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