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:

Write_Mikey_Write!

about 8 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 10 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.

June 26, 2022 at 12:01am

#1034295

The Yeast of Us

Today we're talking about one of my favorite subjects.

Inside the Beer Lab That’s Wrangling Wild Yeasts
Bootleg Biology unlocks new flavors for brewers.

“Yeast has always been the most magical part of brewing,” says Jeff Mello. “It’s the only ingredient that’s still alive when it’s used, and you only have so much control over it.”

Magic, indeed. I'm sure before Leeuwenhoek (not a band name, but the scientist), before we knew what microorganisms really were, it must have seemed magical and/or divine. (Incidentally, Leeuwenhoek came from a family of brewers.

)

Also, with a name like Jeff Mello, if he doesn't call one of the yeast strains "Mello Yello," I shall be sorely disappointed.

In 2013, Mello was a homebrewer who had just left nonprofit fundraising. But a trip to Brussels, Belgium, launched him on a new career in yeast innovation.

Lucky stiff actually getting to go to Belgium.

On his trip, Mello was inspired by local brewers’ use of traditional yeasts, techniques, and equipment. “The more I looked into yeast like Brettanomyces or bacteria like Pediococcus,” he says, mentioning two ingredients familiar to brewers, “it became clear that I wanted to pull back the veil on these wild microbes and figure out new ways they could be used for beer making.”

It's very likely (or so I've heard) that the earliest beers used whatever yeast was floating around. Over time, strains were cultivated (even before the whole microscope thing) and standardized. Belgian beers tend to use a different strain than German ones, accounting for the different taste of each. But there's no reason why other yeasts shouldn't be experimented with.

When he got home, Mello set out a mason jar behind his house in Arlington, Virginia...

Anyone familiar with the DC area should recognize that this is a badly missed opportunity for a pun.

Lacking a scientific background or industry experience, Mello describes many early projects as “total disasters” covered with mold.

Hence the term "experiment." Sometimes experiments fail.

While Mello has been a catalyst in expanding interest in new yeasts, Bootleg Biology’s signature initiative, the Local Yeast Project, is only possible thanks to the wider brewing community.

There are, last I checked, around 10,000 breweries in the US alone (the number keeps changing; I haven't checked in a while and I can't be arsed to now, but that one's in the ballpark and it's got a nice bunch of zeros in it). Oddly, most of them aren't cutthroat competitors; they just like making beer. Problem is, with that kind of market saturation, you still have to find a way to stand out. For a while there, the way to stand out was to get your IBUs up as high as possible. Those were dark days. Fortunately, the overhopped bitter beer craze seems to have died down, and they're messing around with a lot more variety.

And to achieve some of that variety sometimes means using different grains, hops, and yeasts.

Homebrewer Harry Peck, who has contributed to the yeast bank, used Bootleg’s kit to capture two strains from the air in his Garfield, New Jersey, backyard. He then made two beers with the same ingredients—except each featured one of the strains. By comparing them, he learned that one of his strains tasted more of bananas while the other featured a taste of clove.

Generally, banana and clove taste overtones are associated with Belgian beers. However, too much banana flavor can be a sign of poor brewing practice.

Mello also sees the project as a way to help brewers succeed in the industry. He notes that local yeasts, when paired with other local ingredients, can help create a local flavor, giving breweries an edge on competition. “At some point, if you focus on local ingredients, you are creating an identity that can’t be recreated, you know? And to me, that’s a no-brainer way of standing out in that way.”

Remember I said there are thousands of breweries? Well, very few of them distribute nationwide -- ones like Samuel Adams, Sierra Nevada, and New Belgium. There are some considered "regional" that can only be found in a limited area. But the vast majority are local, usually set up by some dude who got into homebrewing and figured he could maybe make some money at it. That, I presume, is the target audience.

This is the case with Place of Origin, a beer made by Harding House, a Nashville brewery that passionately supports local farmers and food traditions. Its brewers challenge themselves to only brew seasonally and include a local ingredient in every batch of beer.

Now, there's something to be said for being able to source ingredients from all over the world. It's one of the few benefits of living in the final days of a declining civilization, along with global beer itself. But I do recognize the benefit of staying local, too.

There's a term generally only used in oenology: terroir. It describes the taste derived from the soil of the grapevines' roots. It's most noticeable in lighter wines; with Vouvray, for example, which is a chenin blanc, the best ones actually have undertones of the ancient soil of le val de Loire. You don't get that as much with beer, or with large-batch wines which are designed for consistency. You're not going to get much terroir from the grains used for the malt, but it seems to me that you can get the same kind of effect with these local yeasts.

Now if only I could get out of here again and try a few...

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