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 5 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 7 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.

April 12, 2024 at 11:00am

#1068587

Coincidences

Gonna have to contradict folk wisdom again: there really is such a thing as coincidence. Cracked has a few examples here:

6 Coincidences Where Different Stuff Developed the Same Exact Way

...except "same exact" is, for most of these, a stretch.

In March 1951, a new comic called Dennis the Menace debuted in the U.S. That very week, a different comic about a different mischievous boy debuted in Britain, and it was also called Dennis the Menace. Both became hugely popular, and neither adapted the other, and neither ripped off the other.

It would be weirder if coincidences never happened. Other things invented in disparate places simultaneously include calculus and the theory of evolution. Those are less coincidence, though, and more about the background having been laid out, setting the stage for ideas that were ready to be invoked.

6. The Tale of Hershey’s and Hershey’s

These two brands didn’t, say, form on different continents with the same name, like how there was one restaurant in Australia called Burger King and was unrelated to the famous burger chain. The two Hershey companies both formed in Pennsylvania, in the same county, within a decade of each other, by unconnected men named Hershey.

Trademark law is complicated and way outside my expertise, but from what I understand, it gets even more complicated when actual peoples' names are involved.

Milton Hershey created the Hershey Chocolate Company in Lancaster, Pennsylvania, in 1884. Jacob Hershey and his four brothers created the Hershey Creamery Company in Lancaster, Pennsylvania, in 1894. The Hershey brothers were not related to Milton Hershey, and they didn’t form a creamery to piggyback off Hershey chocolate. They formed a creamery because they were a family of farmers.

Still, ten years apart, same county? You'd think even back then some lawyer would have advised one of them to change their name.

5. The True Name of Dogs

We’re playing around with exactly how the conversation went down, but the basics are true: The host kept saying “dog,” and Dixon assumed he’d misunderstood the question. Against all likelihood, “dog” really was the Mbabaram word for dogs, though the language shares no other roots with English or with any of the many languages related to English.

There exist other linguistic coincidences, though they escape me at the moment and I can't be arsed to look for them. But given the number of extinct and extant languages in the world, combined with the relatively limited number of sounds a human can make, it would surprise me more if there were no such coincidences.

4. We’re Stuck With Two Different Calories

We measure food using what’s called the large calorie, the dietary calorie or (most confusing of all) Calorie with a capital C. The other type of calorie is the small calorie, or calorie with a lowercase c.

And yet, people act like calories are an ingredient in food, not a measure of energy.

Obviously, we should not use the same word to describe these two different measures, and for a while, people tried calling Calories “kilocalories.” That made far too much sense, so it never caught on.

Pretty sure I've seen kCal in European food labeling, but I could be wrong. In any case, again, this is less coincidence and more sharing a common origin, unlike the dog example above.

Also, it's not that hard to determine which one is meant from context. If it's about stuff you're going to put into your gaping maw, it's Calories. If it's not, it's calories.

3. Nacho, and His Nacho Formula

Nachos were invented by Ignacio Anaya. “Nacho” was his nickname and has traditionally been the nickname of various people named Ignacio. You know that already if you have any friends named Ignacio, if you saw that Jack Black luchador movie or if you watched Better Call Saul.

Okay, gotta admit I didn't know that. I like Jack Black, but haven't seen that movie. Nor have I seen the referenced TV show.

In this example, the coincidence isn't, as one might expect, two dudes named Ignacio coincidentally inventing nachos at the same time. No, it's way cooler than that, at least if you know the slightest bit about chemistry:

The emulsification agent used in nacho cheese today is sodium citrate. Its chemical formula is Na₃C₆H₅O₇. NaCHO. It was destiny.

No, it's coincidence. Stay focused, okay?

2. We Keep Lolling Over Lol

Lol is Dutch for “laugh,” or “fun” or “joke.” This probably derives from an earlier word lallen, which referred to drunken slurring.

While it shouldn't be surprising that there are similar words in English and Dutch, which are, unlike with the dog thing above, related languages, the similarity here does seem to be coincidence in that LOL is an English acronym that happens to mean something similar to what this article claims is a Dutch word for laugh.

But, again, it's not that farfetched.

1. Everything Ends Up Crabs

No, everything really doesn't. Crabs are no more an inevitable end product of evolution than humans or bees. Some scientists a while back pointed out that different evolutionary lineages produced crablike forms, and the popular media ran with that and exaggerated it to "everything ends up crabs."

I can deal with some hyperbole on a comedy site, though.

You see, we have a lot of different types of crab, and they didn’t all form from one crab diverging into different species as evolution progressed. They formed because a bunch of different species all independently evolved into the same basic animal: the crab. Biologists call this process crabification.

Coincidentally (heh), "crabification" is what I call what people do at certain all-you-can-eat seafood buffets.

Often, when different species evolve in some convergent way, we can point to what’s desirable about that trait, which makes everyone naturally acquire it. When a bat and a dolphin each evolve methods to navigate through chirping, that’s weird, but we can easily say why multiple species would evolve echolocation: because it helps them get around. With crabs, we can only speculate on why creatures keep evolving these flat bodies and tiny tails instead of, say, all turning into lobsters.

Fair enough, but it doesn't rise to the level of coincidence if there's something in the marine environment that makes it useful to have a hard shell and pincers with a flattish body plan, even if we can't quite point to what that something is.

This would be like saying "what a coincidence that fish and marine mammals both have fins." Except we know that fins are useful appendages for underwater locomotion, and that underwater locomotion is helpful for underwater survival.

So, honestly, I mostly kept this article in the queue because of the sodium citrate nacho thing. But it's all interesting, as far as I'm concerned.

Because sometimes, there really is such a thing as coincidence.

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