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:

Write_Mikey_Write!

about 11 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 13 hours ago
In response to "Global Domination"

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

3 Like

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.

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.

June 16, 2020 at 12:22am

#985751

Fat Chance

This article is seven years old now, so some of the background science might have changed. Still, I think it's worth a read. Especially since I'm going to use it for the basis of another Win A Merit Badge Mini-Contest!

https://aeon.co/essays/blaming-individuals-for-obesity-may-be-altogether-wrong

The obesity era
As the American people got fatter, so did marmosets, vervet monkeys and mice. The problem may be bigger than any of us

At first glance, this kind of reminds me of the "Mars is getting hotter too hurr durr" climate change denial bullshit. So you'll believe some scientists unquestioningly, but deny others equally unquestioningly? Hm.

But no, this is different from that.

And so the authorities tell us, ever more loudly, that we are fat — disgustingly, world-threateningly fat. We must take ourselves in hand and address our weakness. After all, it’s obvious who is to blame for this frightening global blanket of lipids: it’s us, choosing over and over again, billions of times a day, to eat too much and exercise too little. What else could it be? If you’re overweight, it must be because you are not saying no to sweets and fast food and fried potatoes. It’s because you take elevators and cars and golf carts where your forebears nobly strained their thighs and calves. How could you do this to yourself, and to society?

Sure. And that cancer is because you smoked a cigarette 50 years ago, not because you live next to a covered-up toxic waste dump. And you have a brain tumor because you looked with lust upon a woman, not random genetic or environmental factors.

Moral panic about the depravity of the heavy has seeped into many aspects of life, confusing even the erudite.

Fancy words have seeped into popular articles, confusing even the simpletons.

A report by the consulting firm McKinsey & Co predicted in May 2012 that ‘health and wellness’ would soon become a trillion-dollar global industry. ‘Obesity is expensive in terms of health-care costs,’ it said before adding, with a consultantly chuckle, ‘dealing with it is also a big, fat market.’

So, there's a market incentive to keep people thinking it's All Their Fault and We Have The Answer!

And so we appear to have a public consensus that excess body weight (defined as a Body Mass Index of 25 or above) and obesity (BMI of 30 or above) are consequences of individual choice. It is undoubtedly true that societies are spending vast amounts of time and money on this idea. It is also true that the masters of the universe in business and government seem attracted to it, perhaps because stern self-discipline is how many of them attained their status. What we don’t know is whether the theory is actually correct.

And that's what science is for, if it's not hijacked for government and business purposes like nutrition science has been-- oh, wait.

Now look, I'm saying all this as someone who lost 100 pounds and has kept it off (yes, I'd like to lose 40 more, but other things have taken priority in recent months like attempting to stay relatively sane). "There's something you can do about it" is distinct from "it's all your fault in the first place."

As the Mayor of New York, Michael Bloomberg, recently put it, defending his proposed ban on large cups for sugary drinks: ‘If you want to lose weight, don’t eat. This is not medicine, it’s thermodynamics. If you take in more than you use, you store it.’ (Got that? It’s not complicated medicine, it’s simple physics, the most sciencey science of all.)

Like I said, old article is old. Hopefully by now we've learned our lesson when it comes to elected officials proclaiming science-y things. Hold on a minute while I laugh at my own joke.

In fact, many researchers believe that personal gluttony and laziness cannot be the entire explanation for humanity’s global weight gain. Which means, of course, that they think at least some of the official focus on personal conduct is a waste of time and money. As Richard L Atkinson, Emeritus Professor of Medicine and Nutritional Sciences at the University of Wisconsin and editor of the International Journal of Obesity, put it in 2005: ‘The previous belief of many lay people and health professionals that obesity is simply the result of a lack of willpower and an inability to discipline eating habits is no longer defensible.’

Again, just because some people can do something about it, if only temporarily, doesn't necessarily mean it was all their fault in the first place. I'm eating 1300-1400 calories a day and walking a mile and a half just to stay where I am. Does that sound normal to you? It doesn't to me. To me it sounds like the Red Queen's Race.

But that's an anecdote, not a study. Treat it as such.

Yeah, okay, I've also been drinking more, so that's probably sabotaging things.

Consider, for example, this troublesome fact, reported in 2010 by the biostatistician David B Allison and his co-authors at the University of Alabama in Birmingham: over the past 20 years or more, as the American people were getting fatter, so were America’s marmosets. As were laboratory macaques, chimpanzees, vervet monkeys and mice, as well as domestic dogs, domestic cats, and domestic and feral rats from both rural and urban areas.

Here's another anecdote: I have two cats. One is fat. The other is skinny. They have access to the precise same food and exercise opportunities. Cats are not known for their self-control, nor do they spend endless hours obsessing over their looks.

Obviously, if animals are getting heavier along with us, it can’t just be that they’re eating more Snickers bars and driving to work most days. On the contrary, the trend suggests some widely shared cause, beyond the control of individuals, which is contributing to obesity across many species.

It's clear to me now. The Universe is expanding at an ever-increasing rate (this is a confirmed observation of cosmologists). Therefore, we're increasing our own mass so that our personal gravity can hold us together.

One recent model estimated that eating a mere 30 calories a day more than you use is enough to lead to serious weight gain. Given what each person consumes in a day (1,500 to 2,000 calories in poorer nations; 2,500 to 4,000 in wealthy ones), 30 calories is a trivial amount: by my calculations, that’s just two or three peanut M&Ms. If eliminating that little from the daily diet were enough to prevent weight gain, then people should have no trouble losing a few pounds. Instead, as we know, they find it extremely hard.

I can't help but be reminded of the Dickens quote from the personal finance entry a few days ago. I see a lot of parallels between weight loss and budgeting.

To make sense of all this, the purely thermodynamic model must appeal to complicated indirect effects. The story might go like this: being poor is stressful, and stress makes you eat, and the cheapest food available is the stuff with a lot of ‘empty calories’, therefore poorer people are fatter than the better-off. These wheels-within-wheels are required because the mantra of the thermodynamic model is that ‘a calorie is a calorie is a calorie’: who you are and what you eat are irrelevant to whether you will add fat to your frame. The badness of a ‘bad’ food such as a Cheeto is that it makes calorie intake easier than it would be with broccoli or an apple.

This is, obviously, an attractive model to those who like to make moral judgements when they see a fat person.

We are, of course, surrounded by industrial chemicals.

I am very glad that the author here specified "industrial" chemicals. Everything is a chemical.

In any developed or developing nation there are many compounds in the food chain that seem, at the very least, to be worth studying as possible ‘obesogens’ helping to tip the body’s metabolism towards obesity.

Oh, but we can't focus on that, can we? After all, that would shift blame away from the individual, and that's just not the American Way!

Beatrice Golomb, professor of medicine at the University of California, San Diego, has proposed a long list of candidates — all chemicals that, she has written, disrupt the normal process of energy storage and use in cells. Her suspects include heavy metals in the food supply, chemicals in sunscreens, cleaning products, detergents, cosmetics and the fire retardants that infuse bedclothes and pyjamas.

I bet she has all kinds of fun trying to get grants.

The article goes on to suggest that temperature control and artificial light may also play a role. And now that I think of it, didn't the rise in obesity take off about when central air conditioning became more commonplace?

Then there are the viral and bacteriological hypotheses, which the article also goes into.

These theories are important for a different reason. Their very existence — the fact that they are plausible, with some supporting evidence and suggestions for further research — gives the lie to the notion that obesity is a closed question, on which science has pronounced its final word. It might be that every one of the ‘roads less travelled’ contributes to global obesity; it might be that some do in some places and not in others. The openness of the issue makes it clear that obesity isn’t a simple school physics experiment.

And this is important to note, because none of these proposals are conclusive, from a scientific perspective. But then, neither is the Puritanical hypothesis of "you're a sinner!"

In short, Wells told me via email, ‘We need to understand that we have not yet grasped how to address this situation, but we are increasingly understanding that attributing obesity to personal responsibility is very simplistic.’ Rather than harping on personal responsibility so much, Wells believes, we should be looking at the global economic system, seeking to reform it so that it promotes access to nutritious food for everyone. That is, admittedly, a tall order. But the argument is worth considering, if only as a bracing critique of our individual-responsibility ideology of fatness.

Or we could just blame capitalism.

What are we onlookers — non-activists, non-scientists — to make of these scientific debates? One possible response, of course, is to decide that no obesity policy is possible, because ‘science is undecided’. But this is a moron’s answer: science is never completely decided; it is always in a state of change and self-questioning, and it offers no final answers. There is never a moment in science when all doubts are gone and all questions settled, which is why ‘wait for settled science’ is an argument advanced by industries that want no interference with their status quo.

Also important to understand.

So, now How To Win A Merit Badge.

Above, I suggested, obviously as a joke, that obesity is our way of compensating for the Universe's ever-increasing expansion. So, having at least skimmed this article, and knowing all of these contenders for Why We Are Huge, tell me in the comments:

Why are we huge?

You can be serious, or funny, or both; yes, I tend to appreciate humor, but if there's a gem of dark truth hidden in the humor, even better. As always, you have until midnight WDC time. If I get enough comments, I'll do this again in a few days.

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