Complex Numbers (Book)

Blog Calendar

◄ December ► 2022
S	M	T	W	T	F	S
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28	29	30	31
Archive				RSS

About This Author

Contact Me

Come closer.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 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 4 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.

December 9, 2022 at 12:02am

#1041595

Kanterpoint

Ugh.

How to change your self-limiting beliefs
Let Descartes, Kant and other philosophers help you view the world through a more positive filter and you’ll bloom

I Kant even.

Dammit. I was doing so well, and then I had to go and make a Kant pun. And I was so sure I could refrain. So much for positivity.

Have you ever decided not to go for that job promotion because you believe you’re not qualified enough?

Have you ever considered that maybe you're not qualified enough?

Or avoided asking a neighbour for help because you feel you’d be a nuisance?

Or because he has a shotgun?

Or taken your failure to get what you wanted as confirmation that, yes, your hunch that it was never going to work out was obviously correct?

Feels good to be right, doesn't it?

Philosophy and coaching are a perfect – and underexplored – partnership.

[citation needed]

In fact, you can and should change the beliefs that hold you back. Doing so will make your life go better.

Assertion without evidence. One thing it will do, though, is stop making other people uncomfortable around you. Being a pessimist is kind of like making puns: it's only amusing to the pessimist or punster; everyone else wants to get away. Source: me, a pessimist punster.

What one person views as a job for which she’s underqualified and therefore should not apply, another views as an opportunity that it would be daft not to go for – because, who knows, it might all work out.

Or it might not and you can end up worse off than you were before. "But at least you tried!" "Yeah, but now I'm broke."

When it comes to finding and digging up problematic foundational beliefs, dusting them off, and holding them up to the light for a closer look, philosophers are old hands. It’s at the core of what we do. This process is vividly illustrated in the writing of Descartes, the 17th-century French philosopher. In his essay Meditations on First Philosophy (1641), it occurs to him that everything he knows might turn out to be false, since it’s based on information that initially came to him through his senses, and our senses can sometimes deceive us. He set about rejecting absolutely everything he thought he knew, with the aim of allowing back in only those beliefs that he could be absolutely certain are not mistaken. Eventually – and famously – he arrives at one undeniable truth: that he exists. ‘I think, therefore I am’ expresses Descartes’s observation that, as long as he has thoughts, he can be sure that he exists.

Not going to argue that one, but extending that to being absolutely sure of anything else is kinda shaky. (I will note, as an aside, that when mathematics was re-examined from first principles, Descartes' assertion was essentially the first principle.)

None of us perceives the world as it ‘really is’. The 18th-century philosopher Immanuel Kant distinguished between noumena (things-in-themselves) and phenomena (things as they appear to observers). We can never know noumena, according to Kant; we can know only phenomena.

I just want to take this opportunity to point out that just because a philosopher says something, that doesn't mean it's true. It's possible to build entire edifices of logic on false premises. In this case, if "we can never know noumena," it stands to reason that it doesn't matter if noumena exist or not.

Let me provide this analogy: right now, my chair is keeping me from falling to the floor. Therefore, it exists. A philosopher might argue that the chair doesn't "really" exist because it's made up of materials, which are made up of molecules, which are made of atoms, which are made of protons, neutrons, and electrons, two of which are in turn made up of quarks and whatever. And as science probes deeper, maybe they'll find even more fundamental things. Which are all energy, so the chair therefore can't exist. This is such obvious bullshit that I dismiss it out of hand. It's reminiscent of one of Zeno's several Paradoxes: that you can never get anywhere because you always have to get halfway there first; but by empirical observation, you can indeed get somewhere. It would be like saying that galaxies don't exist because they're made up of stars, when it's bloody damn obvious to anyone who's ever looked through a telescope that they do, in fact, exist. The point being that the reality of my chair is independent of what it's made of, and its existence is very important to my ass.

Kant had his own thoughts about what it is about us that determines the particular spin we put on reality – but we needn’t get into that. Our lesson here can be: we view the world through a filter. That filter comprises our deeply held beliefs, among other things.

On that point, however, I can concur. This is clear if only because it's obvious that different people have different beliefs. That is, if other people exist at all, which is apparently questionable.

Now, okay, I've been ragging on this, but really, some of the things the author writes are worth looking at. You'll need to draw your own conclusions, though. Just as she urges us to examine our fundamental beliefs, I think it's important to think critically about what she's saying.

And one of the things she's saying is that writing can help us examine our own beliefs: "Our ideas – including those we find most compelling – often come to us only semi-formed, and this can disguise their flaws. Simply articulating these beliefs enables us to understand them better, and sometimes reveals that they are just bonkers."

I can't argue with that.

6 Like • Comment • View 4 comments

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.