Complex Numbers (Book)

Blog Calendar

◄ April ► 2023
S	M	T	W	T	F	S
						1
2	3	4	5	6	7	8
9	10	11	12	13	14	15
16	17	18	19	20	21	22
23	24	25	26	27	28	29
30
Archive				RSS

About This Author

Contact Me

Come closer.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

April 20, 2023 at 9:24am

#1048496

Heavy Size

More proof that we're utterly obsessed with size:

What is a black hole’s actual size?
What do we mean by a black hole's size? A photon sphere? The minimal stable orbit? The event horizon? The singularity? Which one is right?

Out there in the Universe, size definitely matters.

Always lead with a subtle dick joke.

Objects that are stable, both microscopically and macroscopically, are described by measurable properties such as mass, volume, electric charge, and spin/angular momentum.

And, lately, political affiliation.

But “size” is a bit of a tricky one, particularly if your object is extremely small.

Snort.

After all, if all the mass and energy that goes into making a black hole inevitably collapses to a central singularity, then what does the concept of “size” even mean?

Obvious jokes aside, I've long wondered how you can measure the size of something that warps spacetime, which makes the concept of "size" very slippery (as if covered with K-Y)

The first thing you have to know about a black hole is this: in terms of its gravitational effects, especially at large distances away from it, a black hole is no different from any other mass.

I've mentioned something like this before, I know, but can't be arsed to find it right now. Black holes have some legitimately scary properties, but if they were going to eat entire galaxies, they'd have done it by now.

After all, we’re taught that black holes have an irresistible gravitational pull, and that they suck any matter that comes too close to their vicinity irrevocably into them. But the truth is that black holes don’t “suck” matter in anymore than any other mass.

I'm not well-versed in the math involved, but there's a region around a black hole within which orbits decay in a predictable way. Outside that region, black holes act like any other mass.

This is complicated by the prevalence of gas, dust, and debris in the vicinity, all of which would cause any orbit to eventually decay.

When a black hole rotates, it no longer has just one meaningful surface that’s a boundary between what can escape and what can’t; instead, there are a number of important boundaries that arise, and many of them can make a claim to being the size of a black hole, depending on what you’re trying to do. From the outside in, let’s go through them.

The rest of the article does just that; it's very interesting, and only a little bit math-oriented, but there's no need for me to comment on it. After all, I just wanted to make the point (pun intended) that even science writers are not above making the occasional dick joke.

8 Like • Comment • View 1 comment

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.