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 3 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 5 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 18, 2021 at 12:01am

#1008685

Just the Facts

Today, just a gentle reminder that we don't always know what we think we know. From Cracked, so it's also amusing.

5 Everyday Science Facts (Nearly Everybody Gets Wrong)

Only it's not limited to science.

We all love science, sure. It's what makes things explode when we want them to, and keeps things from exploding the rest of the time.

Leave it to Cracked to succinctly explain the primary purpose of science.

5. "You Are What You Eat"

And if you eat a lot of a nutrient, you build a whole lot of it into your body. Right? That sounds reasonable enough. But that very last observation is kind of only true with fat, which is something your body likes to store away (and even fat storage is more complicated than that). With other stuff, you have to eat it, but if you eat extra, your body just kind of discards it.

This is demonstrably true, but the article does a pretty good explanation of why it's so.

Anyway, my mom always used that line on me: "You are what you eat." She was a lousy cook and I was a lousy kid, so one time, I said "You're saying I'm garbage?"

That was the last time I said that.

4. "Save The Rain Forest, It's The Earth's Lungs"

Most oxygen doesn't come from trees at all. It comes from algae in the ocean. And you don't get many trendy charities for looking after algae, because no one had fun climbing algae as a kid.

I gotta give 'em this one. Assuming it's as well-researched as the other points, it's something I never looked into, so I didn't know. In any case, there are plenty of other reasons to save the rainforest.

3. "Let's Keep Squeezing Solar Panels Everywhere!"

Having conceded that trees are great, though not in the way some people think, let's also reassure you that we think solar power is great. The tech works, it's advanced enormously recently, and we should build a bunch of new solar arrays. But many people seem to take this to mean that we need a solar panel, like, right there, right where they’re staring right now.

Minor quibble with this one: even deserts have biodiversity and what life exists there is dependent on sunlight. Build giant solar arrays out in Nevada and what happens? Shade on the ground. The ground that's harboring some really delicate life. So if we can put solar panels on things that are already blocking out the ground, like roofs and roads, I have no problem with it. But the article also goes into the economics of such.

2. We Have No Way Of Picturing Extreme Probabilities

This one's a pet annoyance of mine. Large and small numbers are incomprehensible. Hell, to most people, numbers are incomprehensible. It doesn't help that a lot of people don't see the world in terms of probabilities; only in black and white. Either I win a hand of poker, or I don't. Either I win the lottery, or I don't. If I get vaccinated, I could still catch the disease, so why bother? (The answer to that last one is because your chance of catching the disease drops by several orders of magnitude.)

1. None Of Us Can Picture Distances In Space

There's a Douglas Adams quote about how big space is, something about how it's considerably more than even the distance from here to CVS.

The actual quote is "“Space [...] is big. Really big. You just won't believe how vastly hugely mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space.” I first read it when I was fairly young, so I didn't know "chemist" was Brit for "pharmacist," and was confused. What's really confusing is that their word for "peanut" is the same as ours.

In any case, this one is kind of related to #2. Some things are just hard for our earthbound ape minds to comprehend.

4 Like • Comment • View 1 comment

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