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:

Write_Mikey_Write!

about 10 hours ago
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

about 12 hours ago
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

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.

May 11, 2022 at 12:01am

#1032224

Doom and Gloom

Another article that didn't get the memo that the only ones who can fight climate change are us individuals.

Big oil and gas kept a dirty secret for decades. Now they may pay the price

This article is from about a year ago, so I'm going to go out on a limb and guess that they didn't pay the price.

Via an unprecedented wave of lawsuits, America’s petroleum giants face a reckoning for the devastation caused by fossil fuels

And they have an unprecedented army of lawyers, rivaling only that of Disney, to keep them from facing said reckoning.

Coastal cities struggling to keep rising sea levels at bay, midwestern states watching “mega-rains” destroy crops and homes, and fishing communities losing catches to warming waters, are now demanding the oil conglomerates pay damages and take urgent action to reduce further harm from burning fossil fuels.

But I thought the solution was organic, locally-sourced reusable cotton bags.

But, even more strikingly, the nearly two dozen lawsuits are underpinned by accusations that the industry severely aggravated the environmental crisis with a decades-long campaign of lies and deceit to suppress warnings from their own scientists about the impact of fossil fuels on the climate and dupe the American public.

Scientists? Meh. What do they know? Have they built a giant company to slake the public's thirst for energy? No? Then what good are they?

“Things have to get worse for the oil companies,” he added. “Even if they’ve got a pretty good chance of winning the litigation in places, the discovery of pretty clearcut wrong doing – that they knew their product was bad and they were lying to the public – really weakens the industry’s ability to resist legislation and settlements.”

Not going to happen. Corporations, by definition, can do no wrong.

In 1979, an Exxon study said that burning fossil fuels “will cause dramatic environmental effects” in the coming decades.

“The potential problem is great and urgent,” it concluded.

And yet, here we still are, Cassandra.

The urgency of the crisis is not in doubt. A draft United Nations report, leaked last week, warns that the consequences of the climate crisis, including rising seas, intense heat and ecosystem collapse, will fundamentally reshape life on Earth in the coming decades even if fossil fuel emissions are curbed.

But the climate changes all the time. Who are we to think we have the power to change the entire climate? No, better to just let the coastal cities drown like nature intended. So what if a lot of people are displaced? It's their fault for living in places like Miami.

Municipalities such as Imperial Beach, California – the poorest city in San Diego county with a budget less than Exxon chief executive’s annual pay – faces rising waters on three sides without the necessary funding to build protective barriers.

Eh, they're just going to slide into the Pacific come the next big earthquake, anyway.

Farber said cases rooted in claims that the petroleum industry lied have the most promising chance of success.

The only time lying is bad is when a Democrat does it.

Exxon worked alongside Chevron, Shell, BP and smaller oil firms to shift attention away from the growing climate crisis. They funded the industry’s trade body, API, as it drew up a multimillion-dollar plan to ensure that “climate change becomes a non- issue” through disinformation. The plan said “victory will be achieved” when “recognition of uncertainties become part of the ‘conventional wisdom’”.

Conventional wisdom says we should drive and fly more. Gotta keep the economy running.

The legal process is likely to oblige the oil conglomerates to turn over years of internal communications revealing what they knew about climate change, when and how they responded.

Hm. I should have bought stock in shredder manufacturers. Oh well.

Really, there's no need to worry about climate change. Pretty soon the global average temperature will decrease again.

Nuclear winter will have that effect.

Anyway, if it's not obvious, at least from that last bit, my comments here are intended as satire. Still, the time to do something about climate change was 30 years ago. Now, it's too late. I know they keep trying to get us to hope, but it's not working on me. Like I've said before, I'm riding this sucker down with the hot wind in my hair and a big grin on my face.

4 Like • Comment • View 2 comments

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