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 an hour 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 3 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').

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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.

May 31, 2023 at 9:22am

#1050318

It's Not Right

Hey look, we made it to the end of another month. Apropos of nothing:

It’s Time to Ban “Right Turn on Red”
The dangerous maneuver is allowed thanks to a flawed idea about emissions from the 1970s. We don’t need it.

Last time I checked, RToR is the law of the land in the US, everywhere except New York City. That is, in not-NYC, a right on red after stopping is not only permitted but encouraged, unless there is a sign prohibiting it. In NYC, turning right at a red light is illegal, unless there's a sign permitting it.

There are no signs explaining this as you drive into NYC, or out of it. None. It's just something you have to know, and of course NYPD doesn't give a shit and you'll generate some revenue for the city if they catch you.

Even knowing the law, I've made the mistake in that city. No cops around, so no consequences. I was just so used to doing it that I forgot I was in NYC (not hard to do in most areas of the city that aren't Manhattan).

This article talks about the potential ban of the practice in DC. There, it makes more sense to ban it. Driving in DC is way worse than driving in NYC, and what's the point of defaulting to RToR, if something like 75% of your intersections are signed to forbid it? But, in that case, there's only a river along one edge; for most of the city's boundary, you may not know when you're in DC as opposed to Maryland.

Article is from 2022, and honestly, I can't be arsed to check to see if that ban has been implemented, because I only drive in DC under duress.

Regardless, I have a few minor issues with the article.

It’s an obsolete relic of the 1970s oil crisis.

"Obsolete" is an editorial value judgement.

It’s dangerous to pedestrians.

Read further, and you'll see that this statement isn't well-supported by data.

And, if you drive a car in the United States, you likely do it every day.

Unless you live in NYC or, like me, don't drive every day.

It’s time to get rid of right-turn-on-red.

Opinion.

Nothing wrong with stating your opinion, of course. Source: my opinion. But usually, it's clearly labeled as "opinion," not in the "Environment" section and written by someone whose byline says "News writer."

But in the United States, drivers are generally permitted to turn right at a red light, if there’s a big enough gap in the traffic for them to squeeze into. In fact, you’re likely to get honked at if you don’t do it.

Or, as it we are talking about the US, shot at.

I've been honked at for not turning right on red at intersections clearly and plainly labeled No Turn On Red.

Article states RToR isn't a thing in Europe (would be LToR in the UK), but doesn't mention Canada, which I know has RToR. Apparently, it's not a thing in Mexico, but people there do it anyway. I don't know; I've never been to Mexico.

Still, sometimes drivers fail to yield to pedestrians who have the right of way in the intersection. The data on right-turn-on-red crashes might be scarce, but the existing studies suggest that these types of collisions—while rare—frequently involve a pedestrian or cyclist.

Like I said, not well-supported by data. However, from a driving point of view, it makes sense that there would be a hazard. You're at a red light, about to turn right, and you're twisting your neck way to the left to see when the break in traffic comes. When it does, you might peel out into the intersection without really looking to your right—which is, of course, when a pedestrian is just stepping off the curb in front of you.

As for bicycles, some drivers make a game out of hitting them on purpose, anyway.

Last week...

Again, article is from October of last year.

...the Washington, DC, city council voted to ban right-turn-on-red (RTOR) at most city intersections (and to allow cyclists to treat stop signs as yield signs). If Mayor Muriel Bowser signs on and the bill receives Congressional approval, DC will become the second US city after New York not to allow RTOR. DC, which has struggled to curb traffic fatalities, hopes that ending RTOR will make its streets safer for cyclists, pedestrians, and wheelchair users.

As I noted above, a large number of intersections in DC (my 75% number was pulled out of my ass based on experience in that city) are already signed NToR. Banning RToR would likely remove these signs. This seems like a very good way to increase revenue by catching unwary drivers from VA, MD, or points further away, who are used to it being permitted-unless-signed. As with many traffic laws and enforcement policies, they talk about safety, but they're really about revenue generation.

So, why do US cities allow RTOR in the first place? Blame the oil crisis.

The article proceeds to delve more deeply into this idea.

By 1980, RTOR was the law of the land nationally, except in New York City.

I didn't get my driver's license until 1982, incidentally.

Take into account the growing number of hybrid and electric cars, and RTOR makes even less sense. Schultheiss says that electric cars are actually likely to increase the number of RTOR crashes “because their acceleration rates are dramatically quicker than gas powered vehicles.”

This ties in to my "not looking to the right" comment above.

Critics of the DC bill have pointed out the lack of data showing the dangers of RTOR, but many people who don’t use cars know instinctively how dangerous turning vehicles can be.

And yet, they step in front of right-turning stopped vehicles anyway. And even people who do drive can be inattentive as pedestrians.

I'm not trying to victim-blame, here. Sometimes, it can be the driver's responsibility even when it's the pedestrian's fault.

Anyway, you know what would significantly improve pedestrian safety while at the same time reducing idle times at intersections?

Traffic circles.

"But, Waltz, there are traffic circles all over DC; in fact, they're one of the main reasons driving in DC sucks."

True. But those roundabouts are terribly designed, relics of Pierre L'Enfant's original horse-and-buggy design for the city.

Now, it's been a long time since I've been in continental Europe, since before I could legally drive on streets. But one thing I noticed when I was in England, more recently, was the absolute reign of, not QEII (at the time), but roundabouts. I never drove there, but I rode with other people, and circles are pretty much everywhere. So they don't need LToR.

After that rant, you might think I have a strong opinion about RToR in the US. I do not. (I do have strong opinions about the benefits of roundabouts, though.) It doesn't matter to me whether it's allowed or not. What I do care about is that it be consistent, which, given the major exception of NYC, and the proliferation of intersections where it's not allowed and sometimes poorly signed, it never has been.

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