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:

Write_Mikey_Write!

about 9 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.

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Adherennium Plotting something

about 11 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.

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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').

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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.

June 4, 2021 at 12:13am

#1011269

Unproven

Today's article delves into a topic I don't generally like to explore in here, as it can lead to interpersonal arguments. Hell, it's been a cause of more than one war. But it's just too philosophically interesting for me to pass up. So without further disclaimer, an article from the BBC (which is actually reprinting a Conversation article)...

Can physics prove if God exists?

The simple answer here is, as with most headline questions, "no." Physics doesn't "prove" the existence of anything, but rather provides means to describe and predict the behavior of physical phenomena. When Newton figured out how the Earth and Moon orbited each other, and together orbited the sun, he wasn't out to prove the existence of Earth, Moon, or Sun, but to describe their interactions mathematically. We can see these bodies, so their existence is taken as given. When Einstein studied the photoelectric effect, he wasn't out to prove the existence of light. He was attempting to describe some of its observed properties. Whether any of these things exist or not is a matter for philosophers. I'm inclined to take the existence of the planet as a given, since I'm part of it, and to heap scorn on those philosophers who insist it's all an illusion.

But perhaps I'm reading too much into a headline. Even the BBC has to condense an article into a catchy line or question. Let's see what they're actually saying.

The author's introduction, a question posed by a dude from L.A. who shares a name with a late famous British TV host:

I still believed in God (I am now an atheist) when I heard the following question at a seminar, first posed by Einstein, and was stunned by its elegance and depth: "If there is a God who created the entire universe and ALL of its laws of physics, does God follow God's own laws? Or can God supersede his own laws, such as travelling faster than the speed of light and thus being able to be in two different places at the same time?" Could the answer help us prove whether or not God exists or is this where scientific empiricism and religious faith intersect, with NO true answer?

So yeah, just to be crystal clear, there are a lot of levels of quotes there, but again, it's a (probably paraphrased) Einstein quote wrapped in a question from a reader posted in The Conversation and reprinted by the BBC, and now once again quoted here in my blog.

And I'm not going to try to weasel out of this: my personal philosophy, again, is that it's not science's job to "prove" anything, but to describe behavior mathematically. But once more, the article:

I was in lockdown when I received this question and was instantly intrigued. It's no wonder about the timing – tragic events, such as pandemics, often cause us to question the existence of God: if there is a merciful God, why is a catastrophe like this happening?

This is simply a variation of theodicy: if God is good, why is there evil? Theologians and philosophers have been kicking this one around for at least a couple of thousand years, without what you'd call a consensus.

The idea that God might be "bound" by the laws of physics – which also govern chemistry and biology and thus the limits of medical science – was an interesting one to explore.

In my view, you'd first have to explain what you mean by "God." Without that definition, you can come to almost any conclusion.

If God wasn't able to break the laws of physics, she arguably wouldn't be as powerful as you'd expect a supreme being to be. But if she could, why haven't we seen any evidence of the laws of physics ever being broken in the Universe?

This doesn't quite sit right with me, either. It seems a variant of "Could God create an object so heavy that even God couldn't lift it?" I mean, there are neutron stars and black holes, both of which are pretty heavy and still participate in orbital dynamics, so... well, I'm getting off track. Also, if we see "the laws of physics ever being broken," we'd revise "the laws of physics," which aren't actually laws but predictive descriptions. It's happened in the past and hopefully will continue to happen. That's how science gets done.

We learn at school that nothing can travel faster than the speed of light – not even the USS Enterprise in Star Trek when its dilithium crystals are set to max.

Like many of the things we learn at school, this is not precisely the case. No object can accelerate to, or past, the speed of light; no information can exceed that speed; and space itself can, and does, cause distant objects to recede faster than light, which is why we only see part of the Universe. As for the Enterprise, it is of course fiction.

Fortunately, the author goes into this too; I'm not going to quote everything here.

Some argue that we therefore cannot be sure whether the laws of physics could be broken in other cosmic regions – perhaps they are just local, accidental laws. And that leads us on to something even bigger than the Universe.

Again, a quibble about "laws of physics." They wouldn't be "broken" in other regions, but perhaps "different." And again... not laws.

One headache for cosmologists has been the fact that our Universe seems fine-tuned for life to exist. The fundamental particles created in the Big Bang had the correct properties to enable the formation of hydrogen and deuterium – substances which produced the first stars.

This bit is related philosophically to the thing I posted a couple of days ago.

Some argue it's just a lucky coincidence. Others say we shouldn't be surprised to see biofriendly physical laws – they after all produced us, so what else would we see? Some theists, however, argue it points to the existence of a God creating favourable conditions.

But God isn't a valid scientific explanation. The theory of the multiverse, instead, solves the mystery because it allows different universes to have different physical laws. So, it's not surprising that we should happen to see ourselves in one of the few universes that could support life. Of course, you can't disprove the idea that a God may have created the multiverse.

That's the crux of the issue (pun intended) for me -- science isn't there to prove, or disprove, God.

The theory enables something called quantum entanglement: spookily connected particles. If two particles are entangled, you automatically manipulate its partner when you manipulate it, even if they are very far apart and without the two interacting. There are better descriptions of entanglement than the one I give here – but this is simple enough that I can follow it.

It's also simple enough to be misleading. From what I've read, entanglement doesn't allow for the communication of information faster than the speed of light.

So, there is something faster than the speed of light after all: quantum information. This doesn't prove or disprove God, but it can help us think of God in physical terms – maybe as a shower of entangled particles, transferring quantum information back and forth, and so occupying many places at the same time? Even many universes at the same time?

Eh... probably not. See above.

Has this essay come close to answering the questions posed? I suspect not: if you believe in God (as I do), then the idea of God being bound by the laws of physics is nonsense, because God can do everything, even travel faster than light. If you don't believe in God, then the question is equally nonsensical, because there isn't a God and nothing can travel faster than light. Perhaps the question is really one for agnostics, who don't know whether there is a God.

The question may or may not be nonsensical, but I'm pretty sure it's the wrong question in the first place, because it's not one that can really be answered in the framework of science.

This is indeed where science and religion differ. Science requires proof, religious belief requires faith. Scientists don't try to prove or disprove God's existence because they know there isn't an experiment that can ever detect God. And if you believe in God, it doesn't matter what scientists discover about the Universe – any cosmos can be thought of as being consistent with God.

And what I believe is that these descriptions of science and religion are themselves misleading.

The article ends with a quote from Terry Pratchett, which I find amusing for such a philosophical exploration. That alone would have been enough for me to want to share it.

So, to conclude my own take on the topic, I'm not posting this to step on any toes, but simply to present the question and some thoughts about it. I suspect that most religious people, at least here in the West, would balk at the idea of "proof" of God's existence on the grounds that faith doesn't require proof, and most scientifically-oriented people would reject the idea because that's not what science is for.

EDIT:

Oh hell, I nearly forgot with all the big concepts flooding my mind here.

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