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🎁

about 24 hours 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

October 23, 2024 at 8:02am

#1078781

Known Unknowns

Comedy meets science in another Cracked article.

Questions It Looks Like Science Is Never Gonna Be Able to Answer
Those eggheads aren’t so smart after all, are they?

Right, because not knowing everything is the same as knowing nothing.

Science isn’t a discipline known for moving fast, breaking things or shooting first and asking questions later.

Well... kind of. For various definitions of "fast."

4. The Riemann Hypothesis

It’s ridiculously complicated, but let’s just say that according to the hypothesis, proposed by Bernard Riemann, there’s a pattern to the distribution of prime numbers along the number line.

Oh, come on, that's not science; that's math, a separate discipline that will never know everything.

3. Goldbach’s Conjecture

If 160 years seems like a long time, try 282. That’s how long ago Russian mathematician Christian Goldbach theorized that “every even positive integer greater than 3 is the sum of two (not necessarily distinct) primes.”

I invented the principle "never let facts get in the way of a joke," but again... math, not science.

2. The Clarendon Dry Piles

It sounds like a sex act that only exists on Urban Dictionary, but it may be even weirder. It’s two brass bells connected by a pair of batteries called dry piles powered by electrostatic forces. The amount of charge carried between the bells is so small that it appears to be a perpetual motion machine, but the truth is, the batteries are just draining extremely slowly.

So science has, in fact, answered a question: extremely slow drain. At least this one is actually science, not math.

It’s anyone’s guess when the batteries might die, but we’ll probably never know because the hardware will probably break first.

Seems to me that question will eventually be answered, so this section still doesn't meet the promise of the headline.

1. The 500-Year Microbiology Experiment

Also science.

It’s a stunning act of futility and optimism to bet on society surviving another 500 years, but that’s what scientists at the University of Edinburgh did in 2014 when they endeavored to find out if certain strains of dried bacteria can survive five centuries.

This headline sucked. That's another question that will eventually be answered.

That is, if, as the article notes, science is still around in 490 years.

7 Like • Comment • View 2 comments

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