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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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

October 23, 2024 at 8:02am

#1078781

Known Unknowns

Comedy meets science in another Cracked article.

Questions It Looks Like Science Is Never Gonna Be Able to Answer
Those eggheads aren’t so smart after all, are they?

Right, because not knowing everything is the same as knowing nothing.

Science isn’t a discipline known for moving fast, breaking things or shooting first and asking questions later.

Well... kind of. For various definitions of "fast."

4. The Riemann Hypothesis

It’s ridiculously complicated, but let’s just say that according to the hypothesis, proposed by Bernard Riemann, there’s a pattern to the distribution of prime numbers along the number line.

Oh, come on, that's not science; that's math, a separate discipline that will never know everything.

3. Goldbach’s Conjecture

If 160 years seems like a long time, try 282. That’s how long ago Russian mathematician Christian Goldbach theorized that “every even positive integer greater than 3 is the sum of two (not necessarily distinct) primes.”

I invented the principle "never let facts get in the way of a joke," but again... math, not science.

2. The Clarendon Dry Piles

It sounds like a sex act that only exists on Urban Dictionary, but it may be even weirder. It’s two brass bells connected by a pair of batteries called dry piles powered by electrostatic forces. The amount of charge carried between the bells is so small that it appears to be a perpetual motion machine, but the truth is, the batteries are just draining extremely slowly.

So science has, in fact, answered a question: extremely slow drain. At least this one is actually science, not math.

It’s anyone’s guess when the batteries might die, but we’ll probably never know because the hardware will probably break first.

Seems to me that question will eventually be answered, so this section still doesn't meet the promise of the headline.

1. The 500-Year Microbiology Experiment

Also science.

It’s a stunning act of futility and optimism to bet on society surviving another 500 years, but that’s what scientists at the University of Edinburgh did in 2014 when they endeavored to find out if certain strains of dried bacteria can survive five centuries.

This headline sucked. That's another question that will eventually be answered.

That is, if, as the article notes, science is still around in 490 years.

7 Like • Comment • View 2 comments

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