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-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

about 24 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 20, 2024 at 9:56am

#1080216

Powers of Three

From Quanta, and as usual for that source, may not be suitable for numerophobes:

How Base 3 Computing Beats Binary
Long explored but infrequently embraced, base 3 computing may yet find a home in cybersecurity.

There's been talk of quantum computers being used for that, as well. Difference is, from what I understand, quantum computing is still very much in its infancy.

Three, as Schoolhouse Rock! told children of the 1970s, is a magic number.

Thanks. Now I have an earworm.

The number 3 also suggests a different way of counting. Our familiar base 10 decimal system uses the 10 digits from zero to 9. Binary, our digital lingua franca, represents numbers using only the two digits zero and 1.

One of my favorite nerdy jokes: "There are 10 kinds of people in this world: Those who understand binary, and those who don't."

The hallmark feature of ternary notation is that it’s ruthlessly efficient. With two binary bits, you can represent four numbers. Two “trits” — each with three different states — allow you to represent nine different numbers.

"Trits?" No. "Bits" is short for binary digits. If you have trinary digits, using the same convention would yield something way funnier.

It turns out that ternary is the most economical of all possible integer bases for representing big numbers.

The article explains why, or we can just take their word for it. But:

For large numbers, base 3 has a lower radix economy than any other integer base. (Surprisingly, if you allow a base to be any real number, and not just an integer, then the most efficient computational base is the irrational number e.)

That doesn't surprise me at all, except the surprise of being allowed to have a non-integer base number. The irrational number e is 2.71828..., which is closer to 3 than to any other integer.

Despite its natural advantages, base 3 computing never took off, even though many mathematicians marveled at its efficiency. In 1840, an English printer, inventor, banker and self-taught mathematician named Thomas Fowler invented a ternary computing machine to calculate weighted values of taxes and interest.

What pleases mathematicians doesn't usually please the rest of us.

Why didn’t ternary computing catch on? ... Binary was easier to implement.

Thus showing once again that "easier to implement" doesn't always translate to "most efficient to run."

How does this affect us? Well, it doesn't, much. Numbers have to be shifted to decimal notation either way, so we can do things like taxes and budgets. Mostly, I'm just disappointed with "trits."

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