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 2 hours 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.

2 Like

Robert Waltz

about 6 hours ago
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

about 6 hours ago
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

about 6 hours ago
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

a day ago
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

a day ago
In response to "Allo, Ow Arya?"

heightened and delighted attention—Hedention

1 Like

tj in a winter wonderland

Jan 1, 2025 at 1:50pm
In response to "Retraction"

Another interesting entry, thank you. As for the Committee for Bug-Free Food, I have also read some of the information on insects in canned foods. I always wonder how that compares to insects in frozen food, vacuum sealed foods, and irradiated foods. I grew up eating fruits and berries right off the tree/shrub/vine and likely consumed a fair number of insects without even knowing it.

2 Like

Jayne

Jan 1, 2025 at 12:52pm
In response to "Retraction"

Wow, 8 retractions in a month + 60 concerns this year. They should get a new journal tagline:

Chemosphere, the vanity publisher for "science".

"Awesome science journal" - Reddit User RandomDude02342

3 Like

tj in a winter wonderland

Dec 31, 2024 at 4:48pm
In response to "The End of December"

I'll drink to that.

2 Like

Elisa: Snowman Stik

Dec 31, 2024 at 11:07am
In response to "The End of December"

The story behind Dry January (which actually started in the UK) is part typical insufferable but also part fascinating, as it serves as a bit of reckoning with the drinking culture over there. It may be a bandwagon thing, but given how alcohol has legitimately killed people I can see how it's caught on.

3 Like

December 31, 2024 at 9:32am

#1081753

The End of December

Well, here we are at the end of another trip around the sun (or close enough), at a purely arbitrary date on a purely arbitrary calendar.

I had what I consider to be the beginning of the new cycle ten days ago, on the solstice. If anything should mark a transition on what's really a continuum, let it be something real and measurable. But, most of the world uses the Gregorian calendar for recordkeeping and consistency (it is, I'll grant, remarkably good at calculating solar returns), and it's one of the few things most of us share. So we want to impose meaning on December 31 / January 1, fine. Impose it. At least it's usually celebrated with two of my favorite activities: Drinking, and staying up late.

Now, I don't usually make resolutions. I think they're artificial, and set the resolutor (or whatever) up for failure. Besides, it's a bandwagon thing, and I hate bandwagon things.

Another bandwagon thing is the concept of Dry January: the idea of abstaining from ethanol during that calendar month, perhaps in penance for December's overindulgence. This concept legitimately offends me, and I don't get offended easily. In past years, I've simply ignored it and gone on doing what I usually do.

Thing is, contrary to popular belief (that I promote), I don't actually drink every day. Usually once or twice a week. More, perhaps, when I'm on a trip, but only if I'm not subsequently driving. But this New Year's, I've decided to throw personal tradition into the trash and actually make a (gasp) resolution:

Be it hereby resolved that, in protest of the abominable concept of Dry January, Waltz is determined to drink an alcoholic beverage every day during the calendar month of January. This could be a cocktail, a shot of tequila, a dram of scotch, a tot of rum, a bottle of beer, a glass of wine, or the equivalent. More than that is acceptable. Less is not. I do reserve the right to deliberately fail at the resolution in the case of illness or severe injury that requires painkillers, but apart from that, no excuses.

Don't get me wrong: anyone who genuinely wants to stop drinking, temporarily or permanently, as a resolution or otherwise, I wish them well. This is for me. I just have major issues with following a crowd or participating in what's probably little more than abolitionist propaganda.

We'll see how it goes. Knowing me, I'll fail at it like most of us fail at resolutions.

6 Like • Comment • View 3 comments

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