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 an hour 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 3 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.

November 23, 2023 at 10:04am

#1059989

Who?

Inspired again by "Invalid Item"

, because, here in the US, whatever else is commemorated today gets overshadowed by a national holiday, I wanted to highlight a few other observances that fall on November 23 every year, or happen to fall on November 23 this year.

First, let's get the turkey in the room out of the way:

Unthanksgiving Day

and Turkey-Free Thanksgiving.

The first link, there, provides a good bit of detail about Unthanksgiving. In brief, it's also known as The Indigenous Peoples Sunrise Ceremony, and it takes place on Alcatraz on the fourth Thursday in November. Now, I can't claim Native American heritage, and it's against my principles to do anything but sleep at sunrise (especially in that time zone, which, if my luck holds out, I'll enter in a couple of weeks), but I appreciate that it happens. The history of it is absolutely worth reading.

In contrast, no information is provided for the second link, except for its name and that it also coincides with US Thanksgiving. There's another link to follow in there, but I can't be arsed. My only comment on it is that if you're thinking "tofurkey," please stop that.

Some of these "holidays" for the year seem to be made up by kids. In the case of Wolfenoot

, it actually was made up by a kid. It's always on November 23, so it won't always coincide with US Thanksgiving; nor was it, as you might infer from the name, created in the US. According to that link, "Wolfnoot[sic] is a 'celebration of canines, kindness, and humans who embrace both.' Showing kindness to animals, especially to wolves and dogs, is an important part of the day."

I mean, technically, wolves are lupines, but even I tend to cut 7-year-olds some slack when it comes to technicalities. I'm more of a cat person, as you know, but I still like the idea of this holiday. Except for the social media aspects, as detailed at the link.

Another one always observed on November 23 is Eat a Cranberry Day.

I'd love to, but sadly, Dolores O'Riordan

is no longer with us. Nootch!

Confession time: I'd originally thought to make this entire entry about Fibonacci Day.

Like Pi Day (3/14), it's a mathematical pun, and we all know I can't resist those; it's right there in the name of the blog. Instead, I'll keep it brief in the interest of keeping people reading: The Fibonacci Sequence (or Series), which is important in the branch of mathematics known as number theory, begins with 1,1,2,3... And since today is 11/23, well... you know. You get the next number in the series by adding together the previous two; hence, the next number in the series is 5. So maybe I'll raise a glass in celebration at 5pm. Or maybe not; more likely, I'll be napping then.

There are other observances this day, but the last one I want to highlight is of great personal relevance to Me, which makes it the Most Important: Doctor Who Day.

It, too, always takes place on November 23, because that's the Gregorian calendar anniversary of the airing of the first episode, back in 1963. Math-inclined readers may notice that this year's is significant because it's the 60th anniversary. Even more astute readers will note that the first episode aired the day after JFK was assassinated; somewhat famously, that first programme (look, it's British, so I'll use their spelling) was interrupted by ongoing coverage of that event, a big deal even in England.

The personal relevance, though, is only that I was introduced to the show, in syndication here in the US, when I was a kid, and immediately loved it. And still do, in all its regenerations. Not as much as Star Trek, maybe, but one must remember that, at the time, the only Star Trek was reruns of the Original Series, and a few episodes of the possibly non-canonical Animated Series (which is severely underrated). Apart from the movies, new Trek wouldn't come to be until 1987, so I got my SF fix from Doctor Who in the meantime. (The BBC missed a grand opportunity by not making "Doctor Who: The Next Regeneration." Don't look at me like that; they pun more than I do.) Like many people around my age, to me, Tom Baker will always be The Doctor—though I appreciate what other actors brought to the role, especially Tennant.

So of all the things to celebrate today, for me, I think it's going to be that one.

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