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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10 Most Recent Blog Comments:

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

JayNaNoOhNo

Nov 5, 2024 at 1:16pm
In response to "Anomalies"

*Toronto, Canada submits a sternly-worded petition and disputes the center of the universe claim.*

2 Like

October 11, 2022 at 12:05am

#1039041

Soil Degradation

Sadly, "soil degradation" doesn't refer to a BDSM mud fetish. "Journalistic Intentions"

[18+] #6 of 8:

Soil degradation: the problems and how to fix them

"...and why we won't."

A third of the world’s soil is moderately to highly degraded, threatening global food supplies, increasing carbon emissions and foreshadowing mass migration. A change in farming practices has never been more urgent.

Lots of things are urgent now. I'm all out of urgent. You might say I'm det-urgent. Detergent useful for removing soil from garments.

The dirt beneath our feet often goes unnoticed but it is key to sustaining all life on Earth.

I'll just point out that the first organisms on earth existed before soil as we know it, so I wouldn't say "all" life. Just the part we care about. Which, if you think about it, is the problem.

Silvia Pressel, a Museum researcher in the Algae, Fungi and Plants Division, says, 'Soil is full of millions of living organisms that interact with one another. These organisms have a major influence on soil, such as its formation, structure and productivity.'

One must be cautious when using the term "soil." Its definition depends on context and audience. To a farmer, soil is topsoil: that bioactive layer that you plant seeds in. To a civil engineer, soil has a broader definition, including (mostly) purely mineral layers; it's more of a structural thing than biological. To other people, it may have a negative connotation, as in "that diaper was soiled." And then there's the Soil Stradivarius,

which isn't even pronounced the same.

I've worked the land, and I've been a civil engineer. I've even seen a Stradivarius up close (though not that one). Fortunately, I've never had to deal with a dirty diaper.

Anyway, it's clear to me that this article is concerned primarily with the agricultural definition.

Soil degradation describes what happens when the quality of soil declines and diminishes its capacity to support animals and plants.

I'm just including this because for those who won't click on the link, I felt a definition was in order.

It's when the topsoil and nutrients are lost either naturally, such as via wind erosion, or due to human actions, such as poor land management.

Or, often, both. Like you might have heard of the Dust Bowl,

which, contrary to popular belief, wasn't a football game played during a drought.

There are many types of soil around the world. The UK alone has over 700 varieties, such as clay, sand, silt, loam and peat.

Oh yeah, this article is from the UK; I forgot to mention. North America isn't that different in terms of soil types. I learned precise definitions for all of those in engineering school, then promptly forgot them. But I think even an amateur could look at a soil sample and take a fair guess at what its general classification is. In the UK, the most important of these soil types is peat, because it's used in the production of the finest beverage in all the world, single-malt scotch from the island of Islay.

Sadly, as noted, soil is largely non-renewable. At some point, Islay scotch will have to change. That will be a tragedy for all the world, but fortunately it'll happen after I'm converted to soil myself.

Healthy soil has a good combination of soil structure, chemistry, organic matter content, biology and water permeation for its type.

Here in Virginia, there are soil scientists associated with that other state university that evaluate these things for agricultural purposes. I'm sure other localities have similar programs. Thinking of buying some farmland? Get the soil tested first. It may be cheap for a reason.

A typically healthy soil will be teeming with biodiversity and may include a variety of earthworms, 20-30 types of small arachnids, 50-100 species of insects, hundreds of different fungi and thousands of bacteria species.

I knew a guy in middle school who ate dirt. I think we all knew that guy. If you didn't, you were that guy. His favorite saying was, "God made dirt, so dirt won't hurt." Occasionally, I wonder what became of him, and whether he currently supports cannabis legalization.

I'll just point out the logical fallacy there: if God made dirt, then God also made death cap mushrooms. And they definitely hurt.

Nowhere else in the world is nature so densely packed. A teaspoon of soil can contain more organisms than there are humans living on Earth.

[Citation needed] Your gut biome contains trillions of organisms.

Soil plays a vital role in cleaning water. Minerals and microbes filter and buffer potential pollutants, some of which are absorbed by soil particles.

This is how septic fields work. We've also harnessed this in parking lots; biofilters can be used to remove some of the most harmful pollutants from surface runoff.

While soil degradation is a natural process, it can also be caused by human activity. In the last few decades, soil degradation has been sped up by intensive farming practices like deforestation, overgrazing, intensive cultivation, forest fires and construction work.

One of the trickiest parts of designing what we glibly call "improvements" (subdivisions, strip malls, industrial parks, etc.) is dealing with the soil. You take a guess at how deep the topsoil is; that's stripped out and sold to farmers and whatnot, at least the part that isn't used to resurface the green spaces. The rest should balance cut and fill volumes, but at that point we're talking about the engineering definition of "soil." This is further complicated by also having to guess at depth to rock.

These actions disturb soil and leave it vulnerable to wind and water erosion, which damages the complex systems underneath.

We also had to present erosion control plans for approval, and contractors had to follow them. And yet I'm always seeing those measures poorly maintained, with obvious water erosion leaving construction sites.

But it's not just agriculture that is to blame: increasing urbanisation also has a negative impact. The widespread use of tarmac and concrete prevents water from being absorbed into the ground. This results in the death of millions of microorganisms and can lead to water runoff in other areas where it may cause flooding and erosion.

Like that.

Developing a site can vastly increase the runoff from it. While mitigation measures are normally required and implemented, they have their limitations.

Areas that are most likely to be affected are developing countries which usually provide services and materials to middle- and high-income countries. Many of the people who live in low-income countries could be forced to leave their homes in search of safety and fertile lands, resulting in the loss of cultural identity as well as possible economic and political instability in other areas.

Newsflash: this is happening now. Not just due to soil degradation, but they're running out of fresh water. Not just in "developing" countries, but right here in the US.

Many practices can be changed to prevent, and in some cases reverse, soil degradation.

This part of the article is wishful thinking. Most mitigation measures cost money, or reduce a farmer's income, which amounts to the same thing. The economic winners will be the ones who ignore these things and pursue short-term profits. At least in the US, most farming is corporate, and they care about nothing but profit. So, in short, we're doomed.

The global population size is projected to increase from seven billion today to more than nine billion by 2050.

Crop production has risen dramatically over the past few decades due to intensive agricultural practices, but this has had a huge negative impact on the environment and cannot be sustained. In fact, agricultural productivity is now declining because of this, posing a major threat to global food security.

Altering our eating habits and moving towards a plant-based diet is something we can all do to help make a difference.

Or we could have fewer children. I can't say that ecology isn't the main reason I'm childfree, but it's certainly a factor.

Also, fuck this "plant-based diet" crap. Sure, I eat a lot of fruits and vegetables these days, but only as an accompaniment to actual food.

But hey, it's not all doom and gloom. At least nuclear winter will offset global warming.

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