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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Adherennium Plotting something

about 2 hours ago
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For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

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Brandiwyn🎶

Nov 12, 2024 at 10:50am
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).

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Nov 7, 2024 at 10:58am
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Aiva Raine

Nov 5, 2024 at 3:03pm
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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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May 16, 2024 at 8:52am

#1071194

Knead Some Dough?

Jumping back into food today, courtesy of the BBC...

Scientists work to make healthier white bread

Didn't they try to make "healthier white bread" some decades ago? By removing pretty much everything healthy about bread and fortifying the bland, spongy result with "vitamins and minerals?"

I'll grant we know more now. Just saying, sometimes technology gets ahead of itself.

Scientists are trying to create a new type of bread that is just as healthy as wholemeal but looks and tastes like its white counterpart.

Like I said, the article's from the UK, so there may be some phrases lost in translation. Though it's easy enough to figure out what they mean by "wholemeal."

Aimed at lovers of white bread, the project has been funded by the government to improve the health benefits of UK food.

Resisting the temptation to make a lame joke about British food. Instead, I'll just say, "Yeah, good luck on that with Scotland."

The researchers plan to add small amounts of peas, beans and cereals to the bread mix, as well as bran and wheat germ that are normally removed from white flour.

So, take out the bran and wheat germ, and then put them back in?

Another potential for lost in translation: "cereal" can refer to any grain, such as maize (UK for corn), barley (US for corn), rye, spelt... and wheat, which is what gives me pause because isn't the wheat already there?

Dr Catherine Howarth of Aberystwyth University, who is one of its leaders, said scientists had begun to analyse the detailed chemical composition of existing white flour.

It only occurred to me yesterday that some people act like gluten is some sort of food additive, like the TBHQ from the Pop-Tarts entry a few days ago. Which it isn't. It's a naturally-occurring protein in many grains, including wheat.

Of course "natural" doesn't necessarily mean "good," but in this case, my point is that there are no evil corporations going, "I know! Let's save money by throwing in gluten."

It involved adding back smaller quantities of the wheat germ and part of the bran that is taken out in the milling process, she said, as well as adding other grains that are richer in vitamins, minerals and fibre such as quinoa, teff, sorghum and millet.

Today's new word: teff. I'd never heard of it. Turns out it's from East Africa

, and it's a grass seed. Before that freaks anyone out, remember that maize is also technically a grass.

"Using other cereals we can enhance the iron, zinc and vitamin levels and most importantly the fibre content, because white bread has very little fibre, which is so important for good health."

So, Wonder Bread with fiber (look, I'm using US spelling in my commentary, because it's what I'm used to) re-inserted.

I concluded long ago that this quintessential American mass-produced white bread is called that because you have to Wonder what's actually in it.

“Most people know that wholemeal bread is better for you, but a lot of them are put off by the flavour, or because it’s not what they are used to and they are simply not interested,” he said of the challenge.

I've pointed out before that, to me, bread is food; everything else is a condiment. But just like I'm a snob about wine, beer, cheese, and other products of fermentation and/or distillation, I'm also a snob about bread. I like whole wheat bread just fine; I think the flavor usually has more depth. But one thing I can't deal with: whole wheat baguettes. No one's been able to match the deliciousness and texture of a real baguette by using whole grains.

Mr Holister used me as a guinea pig for an early prototype made from a mixture of normal white flour and some added grains and peas.

It was crustier than the white loaves you get from the supermarket - but otherwise looked and tasted like white bread. But there is a lot more work to be done.

To me, "crustier" is a good thing. The brown strip on the edge of your ordinary white bread hardly qualifies as "crust."

White bread has to have minerals and vitamins added to it by law to make up for the goodness that's lost in the refining process.

Pretty sure that's the case in the US, too.

"Critics would say that it is tricking people into improving their diet, but nutritionists would say it doesn’t matter how it’s done - it’s important to get it down people’s throats to improve their health!

"But the jury's out as to whether this new approach will work,” he added.

This critic doesn't say it's tricking people into improving their diet. The only "trick" would be if it wasn't labeled appropriately. No, my concern is that food is meant to be enjoyed, not to be used as medicine.

Still, even the attempt could yield good science, and I can't rail against that.

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