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

10 Most Recent Blog Comments:

tj in a winter wonderland

about 20 hours ago
In response to "Giant Red Flag"

My thoughts were, purchase a bundle of the little red flags they stick in the ground for construction markers and just hand some out to everyone I meet.

2 Like

Leger~

a day ago
In response to "Giant Red Flag"

Pic of the t-shirt on you or it didn't happen.

3 Like

🌖 HuntersMoon

a day ago
In response to "Giant Red Flag"

Elisa: Snowman Stik - Now, let's talk about procrastination...

2 Like

Elisa: Snowman Stik

a day ago
In response to "Giant Red Flag"

Don't worry (or worry). I'll be offering my perspective on this link this year, likely March or April.

2 Like

tj in a winter wonderland

a day ago
In response to "The Year in the Sky"

It all sounds very interesting and hopefully I will be able to keep an eye on the sky and see some of these. It seems more often than not, it's overcast, or has been for past celestial events. I did, however, get a chance to view the Northern Lights around four this morning. The sky cleared sometime during the night and Max, my Husky woke me up to go out and we enjoyed a few minutes of a spectacular light show.

3 Like

Robert Waltz

Jan 3, 2025 at 11:00am
In response to "The Year in the Sky"

🌖 HuntersMoon

, depends on definition. Astronomically, it's a moon. Astrologically, it (along with the Sun) is a planet. Anyway, I read that to be "planetary parade plus Moon," but we can chalk it up to Lifehacker being misleading again.

2 Like

🌖 HuntersMoon

Jan 3, 2025 at 10:52am
In response to "The Year in the Sky"

Regarding the "planetary parade." The moon isn't a planet. It's a moon.

2 Like

Elisa: Snowman Stik

Jan 3, 2025 at 10:46am
In response to "Marguments"

Oddly enough, this entry rolled out the day the squid and I visited Kennedy Space Center. There was actually a Nars rover model we saw with an interesting note on the information plaque: Not funded by taxpayers dollars.

Ultimately, I feel like while the issue of science communication has seen some improvement, a new problem is emerging. Many communicators have not shown an inclination to have public dialogues with groups who question why we're doing things like going back to the moon. There's been no validation of these concerns by science communicators, which is in my opinion not a good look. And lest you think it's a certain demographic with these objections, well, that's a bias that deserves a re-examination thanks to this piece: https://www.cnn.com/2024/01/05/world/peregrine-moon-mission-navajo-nation-object... remains-scn/index.html .

1 Like

tj in a winter wonderland

Jan 2, 2025 at 4:12pm
In response to "Neutral Zone"

I don't tend to think of myself as optimistic or pessimistic, although I do share some of both qualities; I'm realistic.

1 Like

tj in a winter wonderland

Jan 2, 2025 at 3:57pm
In response to "Allo, Ow Arya?"

heightened and delighted attention—Hedention

1 Like

January 3, 2025 at 10:35am

#1081896

The Year in the Sky

Today, instead of picking an article at random, I decided to go with this one from Lifehacker, which plans for the year that's (mostly) still ahead of us. Considering the source, though, you might want to double-check any of these that you might care about. The few that I did check out didn't really track, so I don't trust the rest of it without verification.

Mark These 2025 Celestial Events on Your Calendar
Here's when to look up for full moons, meteor showers, and planetary parades.

No, I'm not going to repeat all of them. That's what the link is for.

Jan. 3–4: Quadrantid meteor shower. The Quadrantids are active from Dec. 28 to Jan. 12 but are expected to peak around 4 a.m. EST on Jan. 4.

Yeah, this is the main reason I jumped this one out of the queue. I've never had much luck with meteor showers—it's always cloudy when the good ones happen, and every time I travel to a darker place to see one, it ends up being a dud—but your experience may vary.

Jan. 13: Wolf Moon. The first full moon of 2025 has extra appeal, as it will pass close to (almost in front of) Mars.

If I had one wish, like from a genie or whatever, I could, of course, wish for world peace. Or an end to homelessness. Or, to be selfish, a billion dollars. But no, if I had one wish, it would be to end this bogus association of moon names with Gregorian calendar months. That's the crusade I'd choose (mainly because I can see how all those other wishes could be twisted to horrific effect, like ending homelessness by disappearing all the homeless).

The Wolf Moon is not defined as the full moon in January, no matter how many websites and "authoritative" sources say it is. It is the first full moon after the northern hemisphere winter solstice. Yes, this year, those happen to be the same thing. But they are not always.

This, as usual, maddened me so much that I almost missed the cool part about Mars. And I also almost missed this:

The red planet will appear to disappear behind the moon at 9:16 p.m. EST and reappear at 10:31 p.m. EST.

I'll give "appear to disappear" a pass, but the last sentence was "close to (almost in front of) Mars" and this one implies it would occult Mars entirely.

This. This is why people don't trust Lifehacker.

So, to break this down a bit:

A Moon/Mars conjunction occurs just about every lunar month. Because most solar system bodies orbit at a slight tilt to each other, eclipses don't happen at every conjunction. This is true for the Sun/Moon conjunction, which is why we don't get a solar eclipse every month, but also for when the Moon appears closest to any given planet. Plus, sometimes, it happens when Mars is in the daytime sky and we can't see it.

Additionally, as with solar eclipses, the timing of the event varies with location, because of parallax. So if you're going to say "9:16 pm EST to 10:31 pm EST," you also have to note the location on Earth where that's true, and it's not "the entirety of places that observe Eastern Time."

Now, a quick glance tells me that other sources give different timings for the eclipse (I guess it should be considered a Martian eclipse), but the January 13 date appears to be correct for both Full Moon and Martian Eclipse. Remember when I said eclipses don't happen at every conjunction? Well, one happening at a Full Moon is remarkable, both for its rarity and spectacle. Of course, it's not happening precisely at the Full Moon (5:27pm EST), but close enough for spectacle.

One wonders if Mars will even be visible against the glare of a Full Moon. I guess we'll find out. Well, other people will find out; something this rare and mars-velous (I couldn't resist) practically guarantees that, wherever I am, the sky will be covered in a thick blanket of clouds.

Nevertheless, I've noted it on my calendar. I've been known to beat the odds before, including during two solar eclipses, and I would very much like to see this.

Well, that took up way more space (pun also intended) than I expected. I'll just throw in some highlights from the rest of the calendar year 2025 skywatch forecast:

February's main event is a planetary parade, when the planets appear to be in one line in Earth's sky. The parade actually begins on Jan. 10 when the Moon joins up with Jupiter and continues through February.

See, again, misleading. The planets will appear to line up, sure, but halfway through that period, the Moon will be on the other side of the sky.

There were a few consecutive nights maybe 30 or so years ago when all the visible planets lined up in one quadrant of the sky. Or at least most of them; it's been 30 years and I barely remember the details. What I do remember is going up to the Blue Ridge Parkway on one of those nights, with some friends and a telescope. It wasn't cloudy. But it was colder than my ex-wife's lawyer's heart. As this is happening in January and February, well, it'll be cold again here.

Saturn will drop off mid-month, but tiny Mercury will be barely visible in the parade on Feb. 28.

And maybe, just maybe, I'll finally be able to see Mercury and know that it's Mercury. I've bitched about that before. I might have seen it at some point in the past, but it's only ever visible just after sunset or just before sunrise, and at those times, the twilight washes out a lot of contextual stars. So I don't know if I've ever actually seen Mercury in the sky.

As a side note, last night, I got treated to a post-sunset very bright crescent Moon not far from a very bight Venus. It was cool. But I expect tonight, the Moon will appear even closer to Venus and it'll look even more awesome.

Provided the clouds don't roll in.

March 14: Total lunar eclipse... Though the total lunar eclipse will be visible around the world, the full 65-minute totality will only happen in the Americas and Antarctica.

Nice for our continent. Unless it's cloudy. I fully expect it to be clear everywhere but Virginia that night, unless I go to, say, Wyoming to see it, in which case it'll be clear everywhere but Wyoming.

Saturn will be at opposition on the night of Sept. 21. Just like Mars in January, this event will show Saturn at its brightest, visible to the naked eye.

To clarify, if Saturn and Earth are in the right alignment such that Saturn appears at night, that planet is always visible to the naked eye. Doubt we'll be able to see the rings without aid, but even binoculars might be enough to make out that distinctive feature of the planet... provided, of course, that the rings aren't edge-on, which I don't know. And if it's not cloudy.

The rest is mostly meteor showers, which, as I said, are cool, but I've never had much luck viewing them. Maybe this year will be different, but I say that every year. And there's a few "supermoons," too, toward the end of the year. Again, though, that just describes a Full Moon near perigee, which happens every year and I think it doesn't deserve all the hype attached to it.

Still, if it gets people to look up, I'll allow it. Just stop with the association of Full Moon names with Gregorian calendar months.

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