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

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.

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