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:

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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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 19, 2024 at 8:22am

#1081306

Five Seconds

Show of hands, please: how many of you are planning on eating over the next week or so?

Huh, that's a lot.

An eating-related article from PopSci:

Is the five-second rule true? Don’t push your luck.
The scientific research on floor food has a clear answer.

I never heard about this "five-second rule" until I was fully an adult. Now, remember, I spent my childhood out in the country, on a farm, and we had our own vegetable garden. The door to the house opened into the kitchen. Clean floors were a "sometimes" thing. But I honestly can't remember what my mom (it was almost always my mom) did if something dropped onto said floor. Probably wouldn't have mattered because I used to pick veggies straight from the garden and eat them. Yes, even carrots. Especially carrots. I wouldn't eat vegetables that she'd cook, but I ate the hell out of raw, dirty-root carrots.

Nurses are always surprised and distrustful when they see "no known allergies" on my chart, but I credit my finely-tuned immune system (quite unscientifically) to eating dirt as a kid.

Anyway, I never believed the five-second rule, and now there's some science to back me up on this.

According to this popular belief, if you drop a piece of food on the floor and pick it up in less than five seconds, then it’s safe to eat. The presumption is that bacteria on the floor don’t have enough time to hitch a ride on the food.

Right, because bacteria are just little animals that take more than five seconds to realize there's a tasty treat above and jump onto it. Snort. No, any bacteria (or other unwanted contamination) hitches a ride on the floor dirt that the dropped food picks up immediately. And I don't care how clean you think your floor is; if it's just been cleaned, there's cleaning agent, which is also not very good for you; and if it hasn't, there's dirt.

In 2003, Jillian Clarke, a senior at the Chicago High School for Agricultural Sciences in Illinois, put the five-second rule to the test.

I will reiterate here that, as a high-schooler, she was younger than I was when I first heard about the five-second rule. Also, we never got to do cool science projects like that in my high school.

Clarke and her coworkers saw that bacteria transferred to food very quickly, even in just five seconds, thus challenging the popular belief.

While this supports my own non-scientific conclusion, one study, performed by a high-school team, is hardly definitive.

A few years later, food scientist Paul Dawson and his students at Clemson University in South Carolina also tested the five-second rule and published their results in the Journal of Applied Microbiology.

Additional studies and replication, now... that starts to move the needle to "definitive."

When they dropped bologna sausage onto a piece of tile contaminated with Salmonella typhimurium, over 99% of the bacteria transferred from the tile to the bologna after just five seconds. The five-second rule was just baloney, Dawson concluded.

One might think that the main reason I saved this article to share was because of the bologna/baloney pun.

One would be correct.

But in 2014, microbiology professor Anthony Hilton and his students at Aston University in the United Kingdom reignited the debate... According to their results (which were shared in a press release but not published in a peer-reviewed journal), the longer a piece of food was in contact with the floor, the more likely it was to contain bacteria. This could be interpreted as evidence in favor of the five-second rule, Hilton noted, but was not conclusive.

Well, maybe UK bacteria are less aggressive.

This prompted food science professor Donald Schaffner and his master’s thesis student, Robyn C. Miranda, at Rutgers University in New Jersey to conduct a rigorous study on the validity of the five-second rule, which they published in the journal Applied and Environmental Microbiology... By analyzing bacterial transfer at <1, 5, 30, and 300 seconds, they found that longer contact times resulted in more transfer but some transfer took place “instantaneously,” after less than 1 second, thus debunking the five-second rule once and for all.

Now that "definitive" needle has moved substantially. But shame on the source for applying "once and for all" to science.

“Based on our studies, the kitchen floor is one of the germiest spots in the house,” Charles P. Gerba, a microbiologist and professor of virology at the University of Arizona, tells Popular Science. Believe it or not, “the kitchen is actually germier than the restroom in the home,” he added.

I get really tired of the "more germs than a toilet seat" scaremongering.

The next time you’re tempted to eat that cookie you just dropped, remember: bacteria move fast.

Or they're hitching a ride on the larger particles that stick to the toast that you inevitably dropped butter-side-down.

Anyway, I'm not sharing this to shame anyone for eating stuff off the floor. You do you, as they say. Just don't make me eat it. My dirt-eating days are long behind me.

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