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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July 20, 2022 at 12:02am

#1035465

Brain Fart

I'm sure most people experience this sort of thing:

What the Tip-of-the-Tongue Phenomenon Says About Cognitive Aging
While word-finding failures can be taken as evidence of memory problems, they may not be harbingers of befuddlement after all.

Still, most of us call it a "brain fart" because it's much shorter and easier to remember. It is extremely unlikely that anyone will ever forget the word "fart," even for a second, because y'all are so attached to the word.

I guess when you're MIT, you can't just use "brain fart," though.

Have you ever had trouble thinking of someone’s name? Perhaps you can even see the face of the person in your mind’s eye, and you would immediately recognize the name if a friend suggested it to you.

I find it sometimes very difficult to recall the names of actors. Even one I've seen hundreds of times, like James Spader. I can only assume that this is because actors take on different names and identities, but sometimes I can't remember those, either. This leads to multiple "You know, that woman. In that movie. The one with the kitchen cabinets? She played a drug dealer or something." moments.

This doesn't happen to me nearly as often with musicians.

In fact, psychologists refer to such experiences as tip-of-the-tongue (TOT) states. But are they really the harbingers of befuddlement that they appear to be?

Having dealt with dementia in both parents, I'm naturally averse to going through it myself. No, there's no genetics involved, but that doesn't mean I won't lose my cognition. Honestly, I'd rather just die suddenly than decline like they did.

Fortunately, I don't see these... TOTs... as being a sign of decline, because, like most people, I've always experienced them on occasion. This is why I'm a writer and not an extemporaneous public speaker; if I'm writing, I can just move on to other things while my subconscious works on finally barfing up the right word or name like a cat with a hairball.

Much like astronomers who study ephemeral phenomena like supernovas, researchers know that TOT states will eventually happen, but not exactly when. This uncertainty has led to two distinctly different ways of investigating TOTs: via naturalistic methods and by experimentally inducing word-finding failures in laboratory settings.

And this is the interesting part: the science.

Diary studies, in which people write down every time they experience a TOT state, allow researchers to assess both frequency and resolution rates. The results suggest that college students experience about one to two TOT states a week, while for people in their 60s and early 70s, the rate is slightly higher. Research participants in their 80s, however, experience TOT states at a rate almost twice as high as college students.

As it is unlikely I'll live that long, that's kind of a... what's the word... relief.

We need to be cautious, however, when interpreting such naturalistic data. It may be the case that older adults, who are more concerned about their memory lapses, will be more likely to record such instances. They may be more conscientious about writing them down, perhaps because their lives are less hectic than those of younger participants.

Which is why you work to minimize confounding variables in scientific studies.

The alternative method for studying word finding is to experimentally induce a TOT state.

When I got to this sentence, I pictured invasive electrodes delivering electric shocks to participants' brains. The reality was a bit of a letdown.

They found that simply giving participants dictionary definitions of uncommon English words would often trigger a word-finding failure. An example from their study was “A navigational instrument used in measuring angular distances, especially the altitude of the sun, moon, and stars at sea.”

Yeah, that would be a bad example for me. Dad was a sailor, and I inherited his sextant, which I keep on a shelf.

But I'm sure others would have worked.

In this study, the participants were often able to provide the desired word without difficulty. On other occasions, the subjects had no idea what word the definition was describing. However, if they found themselves in a TOT state, Brown and McNeill asked them additional questions. The researchers discovered that, while in such a state, people can report partial information about the sought-after word, even as the word itself eludes their grasp.

I've experienced this, too. It's like playing charades with yourself. "Oh, damn, I know this word. Starts with an S. Two syllables. I could make a pun out of it if I remembered it. The pun would have to do with doinking. Ah! That's it! Sextant!"

When given the definition for “sextant,” the participants sometimes responded with “astrolabe” or “compass.” However, they also sometimes offered up words that only sounded like the intended term. The definition for “sextant” also led to responses of “sextet” and “sexton.”

You can almost feel these authors refraining from using salacious examples. Yes, I had to stop to think of the word "salacious."

Incidentally, in case you don't know, it's called that because the name is derived from the Latin for "six," not anything to do with the English word "sex." This, of course, doesn't stop puns from happening. It's almost as reflexive as when someone mentions the name of the seventh planet from the sun.

As with many issues in cognitive aging, we can view the increase in TOT states as a glass half empty or half full. On the one hand, these retrieval failures can be taken as evidence of weakening connections between the meanings of concepts and the words that denote them in long-term memory. It’s also possible that the increase in word-finding problems with age reflects something very different.

Psychologist Donna Dahlgren has argued that the key issue is not one of age but one of knowledge. If older adults typically have more information in long-term memory, then as a consequence they will experience more TOT states. It’s also possible that TOT states are useful: They can serve as a signal to the older adult that the sought-for word is known, even if not currently accessible.

To me, this is the important part of the discussion. While I'm not a fan of people trying to find the positive side of freaking everything, in this case, it would be good to know that having a brain fart doesn't mean your mind is going down the shitter. I say "would be," because that entire second paragraph I just quoted is speculation; clearly, more science needs to be done.

Viewed this way, TOT states might represent not retrieval failures but valuable sources of information. If you are an older adult and still worried about the number of TOT states that you experience, research suggests you might have fewer such episodes if you maintain your aerobic fitness.

Ooooookaaaaay, that exercise thing just came swinging out of nowhere. Oh well. Hopefully, we already knew that; I know that I can write better after some mild exercise. Not excessive exercise, though; that just makes me tired.

Because like everyone else, I'm getting old.

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