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 10 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 12 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.

June 12, 2020 at 12:24am

#985508

Future Tense

Another personal finance article to take a shot at. And I used to respect this source. Not so much anymore.

https://www.kiplinger.com/slideshow/retirement/T047-S014-7-ways-to-sabotage-your...

7 Ways to Sabotage Your Financial Future

I know the url says "slideshow" but it's just one page, at least with my noscript-running browser.

Why would anyone sabotage their own future?

What future?

Here are seven ways you can sabotage your own financial future and work against yourself.

1. Taking advice from someone just because they have a bunch of letters after their name. Oh, wait, that's not actually on there. Let's see what is.

Missing Out on Opportunities (to Save)

By following your FOMO on impulse purchases, you may keep missing out on savings opportunities and significantly decreasing the odds of achieving financial independence.

That's... well, that actually tracks with what I've been saying. Still, it may be worse to completely deny yourself anything "extra" just to sock the money away. Some people can do it. It's possible to do both saving for the future and making the occasional impulse purchase if you plan carefully and already have the basics (food, beer, etc.) taken care of.

Making Big Purchases with Irrevocable Consequences

With a helpful picture of a money pit (boat).

Resale value should always be considered when making big purchases.

I didn't buy my home for resale value. I'm not one of those suburbanite "but so-and-so will make my property value decrease." I *want* my property value to stay relatively low, because I'm paying taxes on the thing, and the more valuable the city thinks it is, the more taxes I pay. I'm not worried about resale value because I don't intend to move.

People be all like, "your house is an investment." No. Your second house is an investment, should you be so fortunate. The house you live in is a home.

And having spent my childhood around boats and boaters, well, no thanks.

Falling Prey to Lifestyle Creep

I live the lifestyle of a creep. Oh, wait, this means something else.

As income goes up, there is a natural desire for some people to increase spending and upgrade their lifestyle.

I don't know about "natural." A lot of that is seeing how other people live and getting into a "keeping up with the Joneses" competition. Some of it is comfort-seeking, though.

By constantly upgrading your lifestyle to keep up with or even outpace income growth, you may find that no matter how much income you earn you always feel squeezed.

One of the sentences in this section made me twitch; I'll let you figure out which one. Anyway, this isn't bad advice as far as it goes. Whenever I'd get a raise, I usually decided beforehand how much of it I'd budget for spending and how much for saving. I've heard "well, you were living on X dollars a year before the raise, and you can keep doing so when you make X+y," but in reality, inflation happens and you have to start spending more just to maintain your standard of living.

Setting the Wrong Priorities

Make sure you’re on track for retirement and financial independence before diverting too much money toward a child’s education or a second home.

I've talked to people who are resigned to working until the day they die. That's anathema to me, so I did something about it. At some point, you're going to become irrelevant and no one will hire you, or you'll be too sick to work, or both. Don't make other people, including your children if you have any, feel obligated to care for you when this happens.

That's actually the point this author makes, so no argument on this one.

Neglecting the Defense

A well-rounded financial plan blends offensive strategies, like investing in stocks, with defensive strategies, such as building an emergency fund and buying adequate insurance coverage.

Yeah, okay, but what to do first? The stock market has become the only way to build wealth long-term, what with interest rates at extreme lows. People see the paltry interest rates on savings and bonds and go, "What's the point?" But at the same time they tend to view the market with distrust, and yet investing in a diverse set of publicly traded corporations over the long term has always worked out in the long run (10-20 years). And if it ever doesn't work out, then we have bigger problems than money.

It used to be there was a middle ground, that, for instance, CDs would pay significantly higher interest than savings accounts, with only the opportunity cost of having your money tied up for a fixed amount of time -- not usually a problem for an emergency fund. But not these days.

Which leads us to

Making Temporary Losses Permanent

Someone told me recently they watched their account daily during the Great Recession in 2008. When their balance plummeted by 50%, they sold all of their investments in stocks and only held cash to avoid losing any more. Worse yet, when the market turned around, they stayed in cash, afraid to invest again for fear of losing more. Instead of avoiding more losses, he simply galvanized that one as a permanent loss of capital.

The biggest mistake people make is buying stocks on margin. Fortunately, that's usually impossible in a retirement account. The second biggest mistake is panic-selling. The thing to do, generally, when the market crashes, is to buy, not sell. This goes against every instinct most people have, so they're frightened of the whole idea.

Market drops, even big drops like the one yesterday, are not only inevitable, but a necessary part of the process. Most people know "buy low, sell high," but what's difficult about that is: how do you know when these points are reached? The answer is, you don't. It has been shown time and time again that there is no way to time the market. Even Warren Buffett, who is probably the greatest investor of all time, can't time the market. Sure, people get lucky sometimes, and you hear about them. And people get drastically unlucky at at other times, and you hear about them too.

But you know what is guaranteed to lose money? Sticking wads of cash under your mattress. If you did that with $1000 in 2010, what would it be worth now?

About $825 in terms of real purchasing power.

If, instead, you put that money into an exchange-traded fund? Well, depending on the fund, and the actual dates, but something like $3000, or $2500 in terms of 2010 purchasing power.

Not to mention that under the mattress is the first place a burglar will look, followed by your sock drawer.

All in all, this turns out not to be terrible advice; I just don't think it goes far enough.

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