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.

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

September 21, 2022 at 12:03am

#1037949

Another Avocado Article

I only had a few articles about avocados in my queue, but for some reason the random number generator decided to lump them all into one month. This month. I think this (from The Guardian) is the last one... for now.

End of the avocado: why chefs are ditching the unsustainable fruit
Give peas a chance – as well as pistachios, fava beans and pumpkin seed paste. These are just some of the ingredients being used to replace one of the world’s most popular fruits

Does the pumpkin seed paste come with pumpkin spice? I'm asking because, though the article is from last November, it's almost the autumn equinox here and now—peak pumpkin spice season. And I'm not immune: I just picked up a six-pack of Atomic Pumpkin Voodoo Ranger, a seasonal beer offering from New Belgium out of Colorado.

On the one hand, they are deliciously creamy, versatile and gloriously Instagrammable.

That last bit means nothing to me, but I suspect they're only photogenic during that fifteen-second period between "too hard" and "rotten."

On the other, they have an enormous carbon footprint for a fruit, require up to 320 litres of water each to grow and “are in such global demand they are becoming unaffordable for people indigenous to the areas they are grown in”, according to Thomasina Miers, the co-founder of the Mexican restaurant chain Wahaca.

On the gripping hand, I'm not sure how much I trust anyone who deliberately misspelled Oaxaca apparently just so gringos won't mangle the pronunciation thereof.

For some time, the chef has struggled to balance the devastating environmental impact of avocado production with her customers’ appetite for guacamole. Now, she thinks she has found the answer: a vibrant, green guacamole-inspired dip, made from fava beans, green chilli, lime and coriander.

Eh, whatever. I rarely eat guac anyway. And when I do, it's spicy enough so I probably wouldn't care if it contained actual avocado or not.

Sucks if you're one of the people for whom coriander tastes like Dawn, though.

The dip – called Wahacamole...

And this is the place where I'd normally close the website window in disgust. But for the sake of my millions of avid followers, I read on. But first I'll note that I would definitely go there if I could just so I could order "whack-a-mole."

In Toronto, the Mexican chef Aldo Camarena recently suggested a guacamole alternative made with courgette and pumpkin seed paste.

That's "zucchini" for Americans. And to be honest, the fava beans one sounds much better.

Last year, the chef Santiago Lastra included a guacamole-style dip made from pistachios and fermented gooseberries on the menu at Kol, his Mexican restaurant in London.

Spain colonizes Mexico. Mexico produces guacamole, then England gets Mexican restaurants and changes the guac. It's the circle of life.

“A few years ago, I was quite well known for my use of avocados in my cooking – so much so that I dedicated a whole Instagram account... to my love for them,” says the vegan cookery writer Bettina Campolucci Bordi.

Wow. I really, really wish I'd stopped reading when my gut screamed at me to do so a couple of paragraphs ago.

She decided to cut back when she moved to the UK, having previously lived in Spain, where she could source avocados locally. “My favourite recipe to date uses British peas instead. I blanche the peas before crushing them and mixing them with plant-based sour cream, salt and pepper, a little grated garlic and a spritz of lemon juice.”

Because before this, mashed peas were unheard of in the UK.

(That's a joke. Mashed peas are the traditional accompaniment to fission chips.)

But avocados are challenging to replace – as are their derivatives, avocado oil and avocado butter, which are important in gluten-free and vegan baking.

Hang on, I gotta find my microscope so I can use it to search for my tiny violin.

Plus, for many fans of the fruit, a dip made from beans, nuts, seeds or vegetables is no more a replacement for guacamole than smashed broad beans on toast (as suggested by Tom Hunt’s recipe for not-avocado on toast) is an alternative to smashed avocado.

Because before this, beans on toast were unheard of in the UK.

Now, just to be clear, I'm not arguing for or against avocados. Personally, I can take them or leave them; the avocado industry could dry up (pun intended) tomorrow and I doubt I'd miss it much. I just thought some of the guac alternatives were creative, so I'm sharing even though the article treats food as a fashion trend rather than what it actually is.

I know I've said this before (sometimes I repeat myself without meaning to because my memory is shit), but one of the few benefits of living on civilization's downhill slide to oblivion is that we can get food from all over. No, I don't give two shits about sourcing food locally; I'm happy to eat tomatoes from California in January, when they're not available from nearby farms, and imported curry from India (for example) all year. But when your only source for a food is a place where it takes all the water to make said food, well, even I will think twice about that.

I'm just not going to take this article's word for that being the case.

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