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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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.

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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.

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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.

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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.

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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').

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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.

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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

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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

November 5, 2021 at 12:05am

#1020862

Solutions

Sometimes, a solution does not exist. Most of the time, though, several solutions exist.

PROMPT November 4^th

You arrive at work today with five donuts and five coffees for the people who are scheduled to be there. But you have a problem. Counting you, six people are there, and all of you love coffee and donuts! How do you and your friends resolve this?

1. Math:

You might have sat there in math class, bored, going, "When will I ever use this crap?" Fortunately for you, I did not.

The fair and egalitarian solution is actually quite simple, but it does require doing some math.

Forget the donuts for the moment and focus on the coffee. You have five coffees. You have six people. Ideally, you'd need to divide the coffee into 30 equal parts, and then recombine them into six, but that presents logistical challenges. So here's the proposed solution:

Find another cup. Since this is an office, you can presume there's a cup, or glass, or some sort of receptacle available. Take 1/6 of each coffee and pour it into the empty cup. Assuming a fairly standard (in the US anyway) 12 ounces of coffee, that means two ounces of each. 2 ounces out of 12 leaves 10 in each and, magically, you end up with 10 ounces in the formerly empty cup as well. Fair, because everyone gets the same amount.

Maybe, like me, you suck at determining how much 2 ounces actually is. Well, one standard US tablespoon is half an ounce. Four tablespoons is therefore 2 ounces. Again, this is an office, so some utensils are certainly present; just use a large spoon; it will be close enough. Unfortunately, the office cup is probably going to be a different shape than the coffee shop cups, so it won't be easy to check, visually, if it contains the same amount of disgustinglife-sustaining fluid -- in which case you'll need to borrow one more cup from the office kitchenette. Empty one of the paper cups into that one, then transfer the contents of the first office cup to the empty one. Then you can visually check if the quantities are close enough, then adjust accordingly.

There's probably a more elegant mathematical solution, but can you really expect us to come up with it before the morning coffee?

With the donuts, it's even simpler to cut 1/6 of each one, leaving 5/6 of a donut for five people, and five 1/6ths of a donut for one person. It may not be ideal to eat pieces of a donut instead of one larger piece of a donut, but hey, it tastes the same and it's all going to the same place.

2. You snooze, you lose:

Who the hell shows up unannounced to work, without anyone texting me to bring six coffee/donut combinations instead of five? Shit, ordering six donuts would earn me fewer weird looks than just ordering five, because six is half a dozen and the donut vendor is just going to assume I'm trying to cut back. Probably cheaper on a per-unit basis, too.

Point being, whoever showed up without notice simply doesn't get a coffee or a donut. Maybe next time they'll figure out they have to text me.

3. Drawing Straws:

Classic survival dilemma solution. Take the coffee stirrers that you no doubt got from the coffee shop. Break them all in half. Break one of them in half again. Take five half-stirrers and one quarter-stirrer and hold them in your hand so only the very end shows. Everyone gets to pick one. Whoever's stuck with the short stirrer is shit out of luck for the day. If you want, you can do separate lots for the coffee and for the donuts.

4. In Reality:

I hate coffee. With five other people, I'm sure at least one of them is trying to lose weight and could be convinced to give up their donut to me in exchange for my coffee. I can even promise to buy them some kale later.

Actually, in reality, this wouldn't happen because I'm retired.

5. Screw You Guys, Then:

Didn't text me to pick up an extra coffee and donut? Screw you guys, I'm drinking all the coffee and eating all the donuts myself. That'll teach ya. Should have paid more attention in math class, or you could have come up with the fair solution, yourself.

Astute readers will note that in exactly zero of the above scenarios do I voluntarily decide to give up all the goodies. I've made peace with that, and you can too.

One-Sentence Movie Review: Last Night in Soho:

It is rare to find a movie outside of one of my preferred genres that is so well-made; it's artistic without being pretentious and tells a great story that's suspenseful and twisty but not gimmicky -- other reviewers might hedge their reviews, knowing that someone out there is going to disagree with them, but that's not a problem for me.

Rating: 5/5

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