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 17 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 19 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.

March 4, 2020 at 12:20am

#977034

Laziness is the Milkman

One of the things that we tend to take for granted -- if we think about it at all, that is -- is the ability of mathematics to model the natural world.

I don't really understand advanced math or physics, but this sort of thing is endlessly fascinating to me. I chose the word "endlessly" deliberately, there; today's link involves infinite, non-periodic patterns.

http://nautil.us/issue/69/patterns/impossible-cookware-and-other-triumphs-of-the...

Impossible Cookware and Other Triumphs of the Penrose Tile
Infinite patterns that never repeat have moved from fantasy to reality.

I first found out about Penrose tiles back in the 80s. I always wanted to cover a floor with them, but I'm entirely too lazy to do the work involved.

In 1974, Roger Penrose, a British mathematician, created a revolutionary set of tiles that could be used to cover an infinite plane in a pattern that never repeats. In 1982, Daniel Shechtman, an Israeli crystallographer, discovered a metallic alloy whose atoms were organized unlike anything ever observed in materials science. Penrose garnered public renown on a scale rarely seen in mathematics. Shechtman won the Nobel Prize. Both scientists defied human intuition and changed our basic understanding of nature’s design, revealing how infinite variation could emerge within a highly ordered environment.

Sometimes, scientists discover something and have to figure out the math. Other times, the pure math comes first, followed by related discoveries. The latter happens quite a lot.

Penrose turned to five-axis symmetry, the pentagon, to create his plane of non-repeating patterns, in part, he has said, because pentagons “are just nice to look at.” What was remarkable about Penrose tiles was that even though he derived his tiles from the lines and angles of pentagons, his shapes left no awkward gaps. They snugged together perfectly, twisting and turning across the plane, always coming close to repetition, but never quite getting there.

The idea that complexity can arise from simplicity is something I've been interested in for a long time. It's right there in the blog intro.

The linked article goes on to explain, in words, what Penrose tiling is. But it helps to see more examples. Here is a more visual explanation.

In fact, Penrose tiles bridged the golden ratio, the math we invent, and the math in the world around us.

As much as I appreciate Nautilus, the promise of the headline is never realized in the text I linked above. It's only mentioned as an aside:

Their irregular atomic configuration, for instance, gives them a low surface energy, which means that not much sticks to them. As a result, quasicrystal coatings have found their way into nonstick cookware. (When Penrose created his novel tiles, there was no reason to think it would have a bearing on crystallography, let alone on frying an egg.)

The point being that all of this stuff seems really esoteric, until you realize that it has real-world, practical applications -- including one of my favorites, an invention that keeps me from having to do too much work.

They say necessity is the mother of invention. I say laziness is the milkman.

4 Like • Comment • View 2 comments

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