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

July 9, 2022 at 12:03am

#1034910

Golden Age

One of the few things all of us still have in common is that we're getting older.

Can you think yourself young?

Them: You're only as old as you feel.

Me: I want to date a 25-year-old.

Them: NOT LIKE THAT.

Guardian link, so UK spellings in the quotes here. Also, it's a cleverly-disguised ad for a book. I've said this before, but that in itself won't stop me from linking it here on a site about writing.

For more than a decade, Paddy Jones has been wowing audiences across the world with her salsa dancing... Jones also happens to be in her mid-80s, making her the world’s oldest acrobatic salsa dancer, according to Guinness World Records.

As an aside, I don't trust Guinness. The story goes that, long ago (mid-20th century), it got its start from that Guinness, the brewery in Ireland. It was marketed as a book to settle pub debates: what's fastest, strongest, biggest, whatever. Drinking culture was baked into it from the very beginning. And yet, now it won't publish any records related to alcohol. This is not the Way. There are other issues with it as well, but in this case, I'm inclined to believe the "oldest acrobatic salsa dancer" thing.

People who see the ageing process as a potential for personal growth tend to enjoy much better health into their 70s, 80s and 90s than people who associate ageing with helplessness and decline, differences that are reflected in their cells’ biological ageing and their overall life span.

That's a bold statement, though again, I'm not going to dispute it here. What I am going to ask is, once again: is it correlation or causation? That is, do people who age well do so because they've convinced themselves that they can? Or do people who age well have that attitude baked in?

The science, however, turns out to be incredibly robust. “There’s just such a solid base of literature now,” says Prof Allyson Brothers at Colorado State University. “There are different labs in different countries using different measurements and different statistical approaches and yet the answer is always the same.”

Yay. Still doesn't answer my question.

But the article then goes on to describe one of the earliest experiments in the field, and, well, it does seem point the arrow of causation from thinking to acting.

Sixty-three per cent made a significant gain on the cognitive tests, for example, compared to just 44% in the control condition. Their vision became sharper, their joints more flexible and their hands more dextrous, as some of the inflammation from their arthritis receded.

That's indeed pretty amazing, but still, I wouldn't call 63% a resounding success. Put another way, it implies that if you were to do the work to "think yourself young," you have less than a 2/3 chance of it working. Put more optimistically, you have nearly a 2/3 chance of it working. (I say this to show that the way we phrase things to ourselves, or in news articles, does make a difference.)

As enticing as these findings might seem, Langer’s was based on a very small sample size. Extraordinary claims need extraordinary evidence and the idea that our mindset could somehow influence our physical ageing is about as extraordinary as scientific theories come.

I'm glad they mention this, because I was about to ask about that, too.

On a study with a much higher sample size:

The participants’ average age at the start of the survey was 63 years old and soon after joining they were asked to give their views on ageing. For example, they were asked to rate their agreement with the statement: “As you get older, you are less useful”. Quite astonishingly, Levy found the average person with a more positive attitude lived on for 22.6 years after the study commenced, while the average person with poorer interpretations of ageing survived for just 15 years.

Well, at least I can't complain that the study was done on university undergrads, the usual target for psychological studies.

“If a previously unidentified virus was found to diminish life expectancy by over seven years, considerable effort would probably be devoted to identifying the cause and implementing a remedy,” Levy and her colleagues wrote. “In the present case, one of the likely causes is known: societally sanctioned denigration of the aged.”

Okay, so now we've switched from people having a good outlook on aging to dealing with society's expectations of aging. While those are obviously related, they're not the same thing.

Still, I can't help but think that our culture (shared with the UK) of celebrating youth, and only giving a shit about what young people say or think, could use a change. Of course I'd say that, though; I'm getting old.

Here's one incredible (to me anyway) result, having dealt with this in my parents, and it being one of my greatest fears about the future:

The most recent findings suggest that age beliefs may play a key role in the development of Alzheimer’s disease. Tracking 4,765 participants over four years, the researchers found that positive expectations of ageing halved the risk of developing the disease, compared to those who saw old age as an inevitable period of decline. Astonishingly, this was even true of people who carried a harmful variant of the APOE gene, which is known to render people more susceptible to the disease. The positive mindset can counteract an inherited misfortune, protecting against the build-up of the toxic plaques and neuronal loss that characterise the disease.

There's obviously no guarantee either way; that's how probability works. Still, a greater chance is a greater chance.

Importantly, however, our age beliefs can also have a direct effect on our physiology. Elderly people who have been primed with negative age stereotypes tend to have higher systolic blood pressure in response to challenges, while those who have seen positive stereotypes demonstrate a more muted reaction.

While I wouldn't take any of these findings as Absolute Truth (it is science, after all, which doesn't deal in that sort of nonsense), having a good attitude about aging is, at the very least, cheaper than drugs or medical treatment, and probably couldn't hurt.

The consequences can even be seen within the nuclei of the individual cells, where our genetic blueprint is stored. Our genes are wrapped tightly in each cell’s chromosomes, which have tiny protective caps, called telomeres, which keep the DNA stable and stop it from becoming frayed and damaged. Telomeres tend to shorten as we age and this reduces their protective abilities and can cause the cell to malfunction. In people with negative age beliefs, that process seems to be accelerated - their cells look biologically older. In those with the positive attitudes, it is much slower - their cells look younger.

This part, to me, is the most unbelievable. I suppose I can think of a mechanism that would produce this effect, though I'm no biologist so I'd have no idea if it's valid or not.

In the meantime, we can try to rethink our perceptions of our own ageing. Various studies show that our mindsets are malleable. By learning to reject fatalistic beliefs and appreciate some of the positive changes that come with age, we may avoid the amplified stress responses that arise from exposure to negative stereotypes and we may be more motivated to exercise our bodies and minds and to embrace new challenges.

You know what one of the best things about aging is? You give less and less of a shit what other people think of you. That's freedom. And freedom from a lot of stress, too.

So, in conclusion, I'm forced to... *choke*... admit... *gag*... that... *cough* *stutter* I...I... was wrong yesterday.

Look, sometimes we're just in a dark place, okay? But it doesn't have to be forever.

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