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:

Turkey DrumStik

about 9 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

October 7, 2018 at 1:27am

#942863

My Last Post on Reality, I Promise - For Now

Kåre เลียม Enga

: I do write fantasy and mostly fiction. It's my way of exploring humanity, a very irrational animal if there ever was one.

Humans are both rational and irrational. What a lot of people miss in discussing the dichotomy is that the opposite of rationality is not emotion (I blame Spock). Emotions are there for a reason, which makes them rational. I'd argue that irrationality can be best defined as acting against our own self-interest - but then we have to get into defining self-interest, which is problematic. A mother protecting her baby can appear to be irrational, but if her actions preserve her legacy (genetic or otherwise - this applies to adoptive parents as well), it can be a successful strategy. People have died or put themselves in harm's way for ideals, as well; standing up for what someone believes in is not necessarily irrational.

Harlow Flick, Right Fielder

: I would add that the burden of proof falls on the person making the claim.

This is absolutely true, in my worldview. It is not possible to prove a negative. Perhaps the most well-known formulation of this is Russell's Teapot

. If you tell me there's an Invisible Hello Kitty guiding my every action, but there's no way for me to sense the presence of said mouthless feline, I'll need something more than your assertion. And yet, I can't prove that the IHK is not there. And I shouldn't have to. (I really hope she's not.)

Which brings me to my original point about reality. Some philosophies claim that there's another reality "beyond" this one, or something similar, and that the world we know is the illusion while this other reality is the "true" reality. Hell, entire religions have been built on such nonsense. Lately, this has manifested itself in the pernicious idea that we're living in a large, complex computer simulation. Even Elon Musk has asserted such, and since he's rather charismatic, people have taken that ball and run with it.

It's unreasonable, irrational, and, most importantly, unverifiable (quantum mechanics is, in contrast, irrational but verifiable). Moreover, it's immaterial (pun intended); it has no bearing on how we treat each other and the very real, verifiable, complex, and sometimes mysterious world around us.

Are there things we don't understand? Of course. Perhaps there always will be. Actually, I hope there always will be, because I don't care to think about what we might do if there were nothing to point our curiosity toward. But that doesn't mean you get to fill in the gaps of your understanding with outrageous claims.

And yes, I consider the idea that there's a supernatural entity who created the universe and takes an interest in who we have sex with to be outrageous. I've no problem with people believing that if it brings them comfort, but when they start trying to make us all believe that way, I start to push back.

I used to work with a devout Mennonite. That's fine; I can get along with just about anybody as long as I avoid the subjects of religion, politics, and whether Kirk or Picard was the better captain. (It was Kirk. Just saying.) One day he (the Mennonite, not Captain Kirk) started showing me optical illusions. You know the ones - some squiggly lines that appear to move; gray dots that aren't really there, but you can see them because of some trick of how the image is arranged; colors that appear different depending on what other colors are around them. You can find examples of all these and more online; I can't be arsed to look them up to link them right now. Anyway, it turns out that what he was trying to tell me was that we can't trust our senses, so we have to trust in God. As revealed through the Bible and interpreted by a particular sect of Shenandoah Valley Christians, in his case.

I looked at these optical illusions and came up with an entirely different conclusion - that since our perceptions are sometimes limited or skewed, we have to seek the reasons why we're seeing stuff that isn't there. Usually it has to do with how our brains interpret the data they're given. And to do this, it requires study, hypotheses, testing, conclusions, and more testing. In other words, science. As a result, we've actually learned stuff about our eyes and brains, which is pretty cool when you think about it - the human brain starting to figure itself out.

And that is what I meant when I said, a few days ago, that what's real is what I can sense; I'd only add the phrase "subject to scientific investigation." Science is specifically designed to overcome the limitations of our senses - telescopes, microscopes, mathematical models, and other tools of science are there to help to reveal the reality that's already there but we can't quite see. Many things that science has discovered are far beyond the perceptions of our common lives, especially when you get into the very big (cosmology), very small (quantum mechanics) or very fast (relativity). And yet they've been tested, and they work.

But sometimes it's wrong and we have to start over. That's okay. I'm comfortable with saying "I don't know." Some people aren't, and they search for, or claim to have found, some Ultimate Truth. And that, in many cases, is what I'd call irrational.

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