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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.
January 30, 2025 at 9:00am January 30, 2025 at 9:00am
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So, anyone know what body part expands up to sixteen times when properly stimulated?
The article, in SciAm, is from 2021, and I don't know if any follow-up science has been done since then.
Our pupils respond to more than just the light. They indicate arousal, interest or mental exhaustion.
Aren't those the same things?
Pupil dilation is even used by the FBI to detect deception.
And I'm pretty sure that's as unreliable as a polygraph.
Now work conducted in our laboratory at the Georgia Institute of Technology suggests that baseline pupil size is closely related to individual differences in intelligence. The larger the pupils, the higher the intelligence, as measured by tests of reasoning, attention and memory.
Now, if that were true, no one could tell that my eyes were blue because the iris would completely disappear. (That's a lie, so my pupils must have changed size.)
In fact, across three studies, we found that the difference in baseline pupil size between people who scored the highest on the cognitive tests and those who scored the lowest was large enough to be detected by the unaided eye.
See what they did there?
We first uncovered this surprising relationship while studying differences in the amount of mental effort people used to complete memory tasks. We used pupil dilations as an indicator of effort, a technique psychologist Daniel Kahneman popularized in the 1960s and 1970s.
I'm pretty sure the theoretical basis of using pupil size as a lie detector rests on the assumption that a lie takes more mental effort than the truth. I'm not convinced that this is the case, at least not for everyone. For example, I find it easier to write fiction than to strain to remember the "true" details of something that I did or happened to me.
The article gets into some of the experimental methodology, then:
We found that a larger baseline pupil size was correlated with greater fluid intelligence, attention control and, to a lesser degree, working memory capacity—indicating a fascinating relationship between the brain and eye.
Fascinating or not, it really shouldn't be too surprising. In some ways, the eye is a direct extension of the brain in a way that other sensory organs are not.
But why does pupil size correlate with intelligence?
To me, that's the real question. What's the mechanism? And what's the connection? (Besides the optic nerve.)
To answer this question, we need to understand what is going on in the brain.
Ah, so you can't answer the question, then.
Okay, I'm joking.
Pupil size is related to activity in the locus coeruleus, a nucleus situated in the upper brain stem with far-reaching neural connections to the rest of the brain.
It's well-known, and not controversial, that the pupil expands/contracts in response to varying light levels. But it doesn't do that on its own; I guess this article is saying that this involuntary response is controlled by that region of the brain.
One hypothesis is that people who have larger pupils at rest have greater regulation of activity by the locus coeruleus, which benefits cognitive performance and resting-state brain function.
I appreciate that this article carefully uses "correlation" and "hypothesis." And that it includes the all-important caveat at the end:
Additional research is needed...
And I'm sure everyone else appreciates that I made it this far without making a cornea joke. |
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