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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

September 27, 2023 at 10:19am

#1056291

Lost in Time

Been a while since I landed on a Cracked article.

6 Secrets We Once Knew But Are Now Lost to History

No, Greek Fire isn't one of these. But rest assured, there are far more than 6.

We have some Russian doodles dating back to the year 1260, showing a six-year-old imagining himself as a knight fighting monsters, but we have other whole years where our records are blank.

D&D is apparently older than we thought.

For example, let’s be honest, can you remember a single thing that happened in 2019?

Only when I look at that year in my blog.

As usual, it's a countdown...

6. What Volcano Blackened the World, Half a Millennium Ago?

It seems like we should know. An eruption that big should have created immediate columns of ash 15 miles tall and would have been audible a thousand miles away. However, we don’t have records from anyone who saw or heard it.

Clearly, it was the one that wiped out Atlantis.

5. What Does pH Mean?

I gotta be honest; this one surprised me. My father, who had an actual degree in chemistry, told me it was "percent hydrogen," and, being younger than 13 at the time, I took that to be the Final Truth on the subject. Once I entered my rebellious years (which never ended), I was too arrogant to ever question that.

You know all about the pH scale if you use quack remedies that promise benefits from balancing the pH in your body.

No, you don't (yes, I know this is what we in comedy circles sometimes refer to as a "joke.")

The H stands for hydrogen — that much, we know. But the p? It doesn’t stand for “percent,” though pH does measure the proportion of hydrogen. Nor, for that matter, does it stand for “proportion.” Many people think it stands for “power,” because pH takes into account exponents, which we sometimes describe using the word power.

Various urban legends say it stands for various foreign words, like puissance, Potenz, pouvoir, potential and pondus.

French, German, back to French, English (which isn't foreign to Cracked readers), and... Latin? The guy who came up with it was Danish; why not look in that language, which I know pretty much nothing about (though it is somewhat related to English and German)?

4. Who Was the Guy on This Coin?

Obviously, "this coin" is pictured in the article.

Silbannacus must have been emperor for a couple months in the year 253. There was enough of a gap in the timeline for him to reign for a little bit. For a short spell in September and October, exactly 1,770 years ago this week, he was the most powerful person in the world. Then he was gone, leaving behind only two coins.

For once, the RNG gave me a recent article, so I need not explain the historical context of the writing I link.

I just want to know if he lasted longer than Liz Truss.

3. What Is This Battery Made Of?

I'd guess... battery stuff?

In 1840, the science of electricity was taking its first faltering steps. Oxford professor Robert Walker bought a curiosity called the Clarendon Dry Pile, which he set up in his lab. It used some kind of chemical reaction to generate electricity, which sent a clapper back and forth between two bells.

Fun fact: we still use "pile" for "battery" sometimes, but from what I understand, the French word is "pile" (pronounced more like "peel.") You can say "battery" in France, but they'll think you're saying "drum" (batterie) which, when you think about it, makes a hell of a lot more sense than English.

It’s been running almost 200 years now, without pause.

And yet, my smartphone battery starts to fail after a month.

The only way to find what’s in the battery is to cut it open. This would end the ringing and may also end the world.

Unless that long-ago inventor managed to stumble upon a way to circumvent the Laws of Thermodynamics, it's not a perpetual motion machine. At some point, perhaps in the distant future when humanity is either dead or living amongst the stars (or both), it'll stop. Then we'll know.

2. What Do Levi’s 501s Mean?

Jeans trivia is its own special branch of knowledge. Did you know that the rivets in denim are essential for holding the fabric together, and aren’t there simply to look cool?

That was true when they were invented. Now, it would be possible to use advanced materials technology to do that with thread alone. But then it wouldn't look nearly as cool.

Did you know that the leather patch that bears the brand logo is called a Jacron?

Okay, I learned something.

And that the tiny upper jeans pocket (the one that fits a single key or an emergency ecstasy pill) was designed to hold your pocket watch?

You do if you read this blog.

And did you know what the “501” in Levi’s 501s stands for? Nope, you didn’t, and as you might now guess, having read this far in the article, no one else today knows that answer either.

You'd think maybe it was just a cool-sounding number, which companies like car makers pull out of thin air all the time, these days. But according to the article (which, in case it's not obvious, shouldn't be taken as the final word on anything, and neither should this blog), there was a whole history behind it. Said history was lost in a fire, much like the contents of the Library of Alexandria.

1. Why Was Ivar the Boneless Boneless?

I'm guessing because his dick didn't work.

Was he missing an arm maybe, or a leg? That sounds reasonable, but it would still leave him with plenty of other bones and may not justify the “boneless” sobriquet. One theory is that he really did have no bones, as we normally know them, but had cartilaginous bones, a condition known as osteogenesis imperfecta. If he did, you’d think every account of his life would have mentioned this fact, long before detailing which parts of Northumbria he raided.

I suppose it's also possible that he was extraordinarily flexible. Physically, anyway.

However, one other theory remains. The name could have been a way of noting that Ivar couldn’t get a boner. We have no record of Ivar having children, and his name may have mocked his impotence.

Or his dick didn't work.

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