Not for the faint of art.
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.
Self-made millionaire: This is the greatest paradox of wealth—and most people fail to recognize it
I think the author and I have different ideas of the meaning of the word "paradox."
*gratuitous photograph of Warren Buffett*
It's remarkable to me how many internet articles about wealth will use stock photos of Warren Buffett. Buffett is undeniably rich - he was once the richest man in the world, and I believe he's still in the top five - but he got that way through diversified investments, not traditional focused wealth-building. And, more importantly, he has nothing to do with the points made in this article; he's known to live a relatively modest lifestyle.
"Oh, I just got back from the grocery store," he said.
After a quick pause, I asked, "Have you ever considered having someone else do your grocery shopping?"
He looked at me like I was crazy. Pay to have someone do his grocery shopping? What kind of elitist would do that?
*raises hand* Me. This kind of elitist right here. The benefits of having groceries delivered, for me, are manifold. Here are a few:
Driving to the grocery store, finding a parking spot, dodging around people, having to come face to face with other people, and trying to make small talk with the cashier are all annoyances to me.
Schlepping the groceries the thirty feet from my car to my door is an even bigger annoyance, especially when I have to make more than one trip.
Done right, it's not much more expensive than doing my own shopping. Instacart Express for the win.
A grocery store is a den of temptation and iniquity. The last time I went into one, because I had to, there was an Oreos display. Resistance was futile. Now, I'm not averse to occasionally indulging, but if I'm shopping every week, well, do the math.
If I go to the store, I will buy beer and wine (VA doesn't do private liquor sales, just fermented beverages). Again, I indulge occasionally, but I have a fine selection of beer and wine (and scotch and gin and tequila and...) here at home, and I don't need to buy any more.
With these last two points, I'm actually saving money in the long run. I buy exactly what I want to buy, no extraneous psychological temptations, and since I'm choosing to eat healthy foods these days, that's a big plus.
Your experience may vary, of course. I understand some people actually enjoy (shudder) going out and mingling with people with all their germs and whatnot, but not this introvert. And some people are better at resisting temptation.
Here was an adult earning $750,000 a year, but behaving as if he still earns $50,000.
That's generally how you save money. It doesn't matter how much money you make; if you spend it all, you're boned.
This is a huge paradox of earning more money: Many people claim they value time over money, but if you look at their calendars, you'll find that the opposite is true.
Again, not a "paradox." Cognitive dissonance, maybe.
I used to scoff at people who flew first-class and think, Why would anyone spend an insane amount of money something so pointless? We're all getting to the same destination.
Because I actually fit in the seats, and because there's nothing more satisfying than sipping on your first complementary gin and tonic whilst watching all the harried passengers struggling past to get to steerage.
If you're making more money than ever, aim to save at least one hour per week. Think about all the responsibilities that you hate (e.g., doing the laundry, grocery shopping, managing your finances) where there are great solutions available to outsource or systematize the work.
Despite my snark above, I think the author makes good points. For myself, I'd rather spend a lot of time up front automating a task - or contracting someone else to do it, like I do with my lawn care - than do the task myself, especially if the task is repetitive. That's how I got good at spreadsheets.