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:

Wordsmitty ✍️

about 2 hours ago
In response to "Losing Weight Is Easy"

This brings up the old question of whether your weight changes when you inhale a helium balloon to make your voice higher.

Maybe this has to do with the difference of a mug of beer vs a tankard of ale.

The universe has so many weighty/massive questions.

2 Like

Brandiwyn🎶

about 3 hours ago
In response to "That's the Spirit"

Maybe you’d go with “Thunder” because that one’s so terrible that only divine intervention can explain its success. Wow. Missed opportunity, here. How could you let this go without a shoutout to mighty Thor, son of the Allfather, wielder of Mjolnir, He with the Long Forehead? (Said in the voice of the Soup Nazi): "No Valhalla for YOU!"

Brandiwyn🎶

about 3 hours ago
In response to "Losing Weight Is Easy"

Has anyone ever told you how annoyingly pedantic you are?

Damn your pedantic ways. I was hoping to justify a vacation to Venus (which I still argue would be a much better holiday destination than fr-fr-freezing Mars, Elon. Ugh.) Now I guess I'll have to settle for the Cancun plans we already made, since my mass (and waist diameter, and thigh diameter) will be exactly the same regardless.

2 Like

tj-turkey-jobble-jobble-hard-J

about 23 hours ago
In response to "Revisited: "A Frank Discussion""

I became disgruntled with meat sold in local grocery stores; half the weight is water, the taste is bland, and it's "use by" date is usually one day away. I found a small town butcher/locker plant that sells locally raised meats and buy in bulk so it's actually fresher, cheaper, and better tasting. Recently, I decided to try my hand at making our own sausage as well. It's a bit of work, but I think it's worth the trouble.

1 Like

Turkey DrumStik

Nov 16, 2024 at 9:24am
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

November 18, 2024 at 10:25am

#1080127

Losing Weight Is Easy

Big Think offers up another way for people to annoy pedants, and vice-versa.

It’s important that weight and mass are not the same
Here on Earth, we commonly use terms like weight (in pounds) and mass (in kilograms) as though they’re interchangeable. They’re not.

By "we," I suppose they mostly mean "Americans." But even people in countries that use SI units will use kg as if it's a unit of weight. This is fine, as far as I'm concerned, because damn near 100% of us live on the Earth's surface. And while there are gravity variations due to latitude, elevation, or anomaly (such as the one from "Anomalies"

a couple of weeks ago), they mostly don't make much difference unless you're a scientist who requires greater precision.

Still, as a pedant, I think it's important to know the difference. Also, as a pedant, I think it's important to keep it to oneself if one does not wish to be uninvited from social gatherings for being a pedant.

Since this is my blog, I'm making an exception here.

Conventionally, here on the surface of the Earth, we can convert between the two using only a minimal amount of effort: 1 kilogram is 2.205 pounds, and vice versa, 2.205 pounds converts to 1 kilogram. Going back-and-forth requires only multiplication or division, which seems easy enough.

"Easy?" Have you met people? Some of them break out in hives if you ask them to add 10% to something.

Incidentally, I don't usually bother with the 0.005. For most practical purposes, 2.2 is close enough and much easier to do head-math with.

A kilogram is an example of mass, not of weight, while a pound is an example of a weight, not of a mass. It’s only here on the surface of the Earth, where we’re at rest relative to the rotating Earth, that these two concepts can rightfully be used interchangeably.

I mean, as long as we're being pedantic, it's a really stupendously big universe and it wouldn't surprise me if there are other planets with the same acceleration due to gravity as that of the Earth.

But for nearly 100% of the volume of the universe, weight is irrelevant and only mass matters (pun absolutely intended).

If you release an object from rest and allow it to fall, it falls straight down, accelerating at a constant rate. It gains speed directly proportional to the amount of time that it’s been falling, and the distance it covers is proportional to the amount of time squared that the object has been falling.

Again, if you're going to get technical about it, you'd add "without air resistance."

This phenomenon, however, appears to not depend on mass or weight. A light object will fall just as quickly as a heavy object, especially if air resistance isn’t a factor.

There it is.

Someone did that on the Moon,

incidentally. This proved two things: 1) Physics is right; 2) at least that particular Moon expedition wasn't faked on an Earthbound sound stage.

The article proceeds to go into a lengthy (and weighty) discussion of the differences between scales and balances, and how the former measures weight while the latter measures mass. It's remarkably math-light.

Then:

We can sum up the difference succinctly: your mass is an inherent quality of the atoms that make up your body, but your weight is dependent on how those atoms accelerate under the influence of all the factors and forces acting on it.

Which I suppose is the main point of the article; everything else exists to support it.

There are plenty of physics textbooks (and physics teachers) that ignore this difference, and simply state that your weight, W, always obeys the equation W = mg. This is incorrect; it is only true when you are at rest on the surface of the Earth.

There's a popular trick question: "Which weighs more, a pound of feathers or a pound of lead?" It's a trick for several reasons.

We often talk about “watching our weight” or “trying to lose weight,” but if that was truly your goal, you could simply go to a higher elevation, move to a different planet, or even get into an elevator and wait for the door to close after you hit the “down” button.

Yes, "simply... move to a different planet."

Sounds like a good idea for other reasons, these days.

1 Like • Comment • View 2 comments

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