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:

Soldier_Mike

about 8 hours ago
In response to "Story Time"

Causation vs correlation. Reminds me of the phrases we've all seen here on site with respect to comma usage (e.g. Come down and eat, children! vs Come down and eat children!).

1 Like

tj wanderlust-words-in-motion

about 19 hours ago
In response to "Just Teaming"

What about "teamMe"!

When I hear "team player" I also hear the boss expecting everyone to give 110%, then going back to the office instead of pitching it.

2 Like

🌖 HuntersMoon

about 22 hours ago
In response to "Just Teaming"

3 Like

tj wanderlust-words-in-motion

a day ago
In response to "One Fine Thing"

Soldier_Mike

recently I asked on of my grandchildren if she new what an octothorpe was (she uses hashtag) and after a minute of deep thought replied, "A mythical creature."

1 Like

Soldier_Mike

a day ago
In response to "One Fine Thing"

Regarding your Exclaimer!, I never heard the term octothorpe before; it was, as you point out, either pound or number (sign) in my experience. As an aside, it's only visually similar to, not identical to and interchangeable with, the musical sharp sign. The octothorpe's left-to-right lines are horizontal, while the similar lines on the sharp sign are slightly angled from lower left to upper right (# vs ♯).

2 Like

tj wanderlust-words-in-motion

a day ago
In response to "Story Time"

It is amazing how you can give some people solid facts and they will doubt them, but make up a story filled with fallacies and they will believe it.

1 Like

Elisa, Stik of Clubs

a day ago
In response to "Story Time"

I might do my own commentary on this article, but for now, I will tackle one part.

The science of vaccinations and immunization schedules also seemed impenetrable. “I was very conflicted, and felt like: ‘I don’t know any of this stuff, I didn’t go to school for this,’” she says.

I struggle with this a lot. I really do. Part of me struggles because while science wasn't my strongest subject in school, I still did reasonably well in the various fields I did study. Biology happened to be my strong suit there, although I strangely managed to hold my own when I took intro to physics. That said, in my science education we didn't go over specific vaccination schedules or anything like that. However, between learning how vaccines worked and getting a halfway decent statistics education (a college semester and several introductory lessons in high school) I could draw the connection of vaccination schedules. So while people may think they didn't go to school for things like this...they might have without knowing it!

The other part of me wonders about the effectiveness of science communicators. This field admittedly is still pretty new (only gaining steam in the last decade). On the other hand, how effective are they in getting laypeople to truly pay attention? I feel like some of their work tends to kind of preach to the choir. I wonder if many of these communicators lean more on the science part rather than the communicating, and that's where things go awry.

4 Like

tj wanderlust-words-in-motion

Mar 6, 2025 at 1:59pm
In response to "Shhhhhhh"

I may have a slight allergy to delusional people; they give me a headache.

1 Like

Wordsmitty ✍️

Mar 6, 2025 at 12:40pm
In response to "Shhhhhhh"

I drove by that several decades back either coming or going to a vacation somewhere beyond. It's so strict that even birds couldn't tweet there. (someone had to say it)

2 Like

BIG BAD WOLF is Merry

Mar 6, 2025 at 9:29am
In response to "Shhhhhhh"

Always someone complaining about something, annoyingly.

1 Like

March 7, 2025 at 8:44am

#1084950

Story Time

Articles like this one from The Guardian give this old pessimist some hope.

Formerly anti-vax parents on how they changed their minds: ‘I really made a mistake’

Strength of character, you see, isn't about clinging stubbornly to one's beliefs. It's about being open to changing one's mind due to overwhelming evidence, especially when going against the collective will of your tribe. Admitting you were wrong demonstrates good character, as well.

Researchers believe personal stories are more persuasive than facts alone in addressing vaccine skepticism

This doesn't surprise me. This shouldn't surprise anyone who writes, be it fiction, nonfiction, or anything in between (tabloid journalism or political speeches, e.g.) And that goes for almost any subject. If you're told that eleven million people died as the result of actions taken by Nazi Germany, you might register that as a fairly large number; if you read the Diary of Anne Frank and learn its associated stories, the horror might actually hit home.

There's a quote attributed to Stalin related to this: "A single death is a tragedy, a million deaths is a statistic." There's no evidence that Stalin ever actually said or wrote that, and plenty of evidence of similar sentiments from earlier in history.

Stalin or not, there is a darkness lurking within the persuasive power of anecdotal storytelling. As with almost everything, its power can be used for good or evil, much as a hammer can be used to create or destroy. I'll get to that shortly, after I look at the article a bit more.

When Nikki Hill Johnson’s first daughter was born in 2012, Johnson didn’t hesitate to take her to the doctor for routine infant immunizations.

And so the anecdote begins.

Soon after the birth, South Carolina-based Johnson, now 42, joined a fitness- and nutrition-oriented multilevel marketing company (MLM).

Speaking of Stalin, wow, that's one big giant red flag.

There, she encountered a colleague who made her question the safety of vaccines.

This does not surprise me, either. Once you hold one set of beliefs without evidence (in this case, that MLMs benefit anyone but the founders thereof), you invite other sets of beliefs without evidence into your mind (in this case, that vaccines aren't safe).

I'll take that even farther. Not only are these beliefs without evidence, but they are beliefs with overwhelming evidence to the contrary. This isn't like religion, where it's not possible for realists to prove that the supernatural doesn't exist (only that it doesn't have to exist); this is willful blindness to overwhelming evidence.

“I remember someone in the community talking about their child who had been ‘vaccine injured’,” Johnson recalls. “It sparked a fear in me.”

Here we come to the dark side of anecdotes: they can be used to sow doubt and fear just as much as they can be used to elicit compassion. Hearing a story like that, believing it to be true, taps right in to our worst nightmares. It's like those attempts to keep kids from doing drugs: "John tried heroin once. Once. Then his arm fell off, his eyes exploded, and he died in agony." Except those are well-meaning, whereas vaccine rejection (I will not call it skepticism, because skepticism isn't about blindly rejecting a premise, but about looking at it rationally) actually does kill.

There also needs to be a better understanding of the difference between correlation and causation. "Jane ventured into the park and died. Don't go into the park." Except that it wasn't anything in the park that killed her; it was a heart attack. So it must have been exercise that actually killed her. See? Exercise can kill.

The science of vaccinations and immunization schedules also seemed impenetrable. “I was very conflicted, and felt like: ‘I don’t know any of this stuff, I didn’t go to school for this,’” she says.

And I have to admit: That, I understand. Science in general is beyond the grasp of most people. They don't understand it any more than I understand what dolphins are saying to each other. Just to be clear, I'm not ragging on that. I can't understand all of it, either. No one can, not even other scientists. Those who aren't lawyers can't comprehend the nuances of law. Those who aren't archaeologists might think Indiana Jones is true to life (spoiler: it is not).

The answer, however, is not to "do your own research" by finding and believing those anecdotes that either speak to your existing biases, or claim to be able to calm your fears.

Johnson joined “crunchy” Facebook groups full of moms concerned about potentially harmful substances affecting their children, and started taking advice from a trusted anti-vax family member who seemed particularly credible because she was a nurse.

And, of course, there's that minefield.

Anyway, there's a lot more to the article. It may or may not help. But at least it's on the right path.

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