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!

a day 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

a day 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.

March 2, 2019 at 12:22am

#953528

I Know

Today's "you're doing it wrong" entry applies - as so many such articles do - to parents.

https://www.theatlantic.com/education/archive/2015/06/the-s-word/397205/

At whatever age smart people develop the idea that they are smart, they also tend to develop vulnerability around relinquishing that label. So the difference between telling a kid “You did a great job” and “You are smart” isn’t subtle. That is, at least, according to one growing movement in education and parenting that advocates for retirement of “the S word.”

I think that brings the number of words you're not supposed to say in front of your kids up to 3,255. Pretty soon you won't be able to say anything to them.

The group most damaged by fixed-mindset thinking is high-achieving girls, Boaler argues, because it’s girls who are told by society that they probably won’t be as good as boys at math and science.

So, wait - follow the logic here - if you tell someone she's "smart," it's because she's a high achiever. But then - somehow - this turns them into underachievers because reasons?

“When we give kids the message that mistakes are good, that successful people make mistakes, it can change their entire trajectory,” Boaler said. 100 percent is not an ideal score. When kids come home from school and announce that they got everything right on their school work, Dweck advises parents to offer some sympathy: Oh, I’m sorry you didn't get the chance to learn.

And, meanwhile, those who consistently get 100 percent scores will go on to the likes of Harvard and MIT, leaving those who made mistakes in the dust.

When kids get the idea that they “aren’t math people,” they start a downward trajectory, and their career options shrink immediately and substantially. There is also the common idea of a wall in math: People learn math until they hit a wall where they just can’t keep up. That wall may be trigonometry, and it may be advanced calculus, and it may be calculating a tip.

I can agree that math is important. But what I see holding people back is the repeated question: "How am I going to use this stuff in the real world?" And that, it seems to me, more than anything else, keeps people from learning. I hate that question. It's why I could never be a teacher and one reason I never wanted to become a parent: ask me that once, and I will give up on teaching you, because you've obviously (in my mind) given up on yourself.

All knowledge is useful. Some, admittedly, is more useful than others. But the more you learn, the more connections you can make, and the more you can at least read shit like this critically.

Full disclosure: I was called "smart" from a young age. I also made plenty of mistakes, and learned from them. But I also learned from teachers. I also learned that "smart" isn't valued in our society. And that - not mere words - is what we have to change if we're ever going to amount to anything.

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