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:

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

October 18, 2022 at 12:01am

#1039361

The Once and Future Viking

In today's "we were wrong all along" news...

Goodbye to the Vikings
The term ‘Viking’ as it is commonly used is misleading, warping our perception of the Middle Ages. It should be retired.

That yipping noise you just heard was the collective barking of Minnesota sportsball fans.

There was no such thing as a ‘Viking’ in the medieval period. Use of the term emerged in the 19th century. The word wicing occurred in Old English and víkingr in Old Icelandic, but were used very differently, to mean something like ‘pirate’.

Which explains much about a certain cruise line and its pricing structure.

In Old Icelandic víkingr could be applied to any pirate regardless of where they came from or when, or what language they spoke; they might be Estonians or Saracens, for example. It is also noteworthy that it is almost never used to describe the people who we today call ‘Vikings’. Many of the men labelled ‘Vikings’ in textbooks and popular histories were warriors led by kings on military expeditions with clear political objectives, such as the Great Heathen Army that fought Alfred the Great or the Norwegian force that accompanied Harald Hardrada to his death at Stamford Bridge in 1066. Calling such people ‘Vikings’ would be like calling 18th century British, French or Dutch naval officers ‘pirates’ simply because they wore vaguely similar hats and sailed vaguely similar ships to Blackbeard.

Have you seen what 18th century British, French, and Dutch naval officers got up to? That's not far off the mark.

The final development, the ‘ethnicisation’ of the word that allows the use of terms such as ‘Viking farms’, ‘Viking towns’ and ‘Viking women and children’, is much more recent and has gradually crept up since the Second World War. This is insidious; by linking military prowess and savagery to an entire ethnic group, it encourages its appropriation by racial supremacists.

Right, because when I think of military prowess and savagery, I think of Scandinavians and not a certain Mongolian.

On the other hand, my ex-wife is of Norwegian descent, so I get the savagery part.

The issue with the term is not merely semantic. This conception of ‘the Vikings’ seriously distorts our understanding of European history. We have tended to group almost all Scandinavian activity between the 790s and the mid-11th century together under the ‘Viking’ label, creating a distinct ‘Viking Age’ and an imagined ‘Viking’ culture and identity. The evidence, however, does not support this analysis.

The article goes on, of course, to explain said evidence, not all of which I'm going to repeat here.

Irish, English and Frankish chronicles generally refer to Scandinavian aggressors as ‘heathens’ and this, rather than any ethnic identity, seems to have been what struck the victims of these attacks as significant. The 793 raid on Lindisfarne, often said to herald the ‘Viking Age’, is described in the Anglo-Saxon Chronicle thus: ‘The ravages of heathen men miserably destroyed God’s church on Lindisfarne with plunder and slaughter.’

Hey, Lindisfarne again (from a couple of entries ago).

The construct of the ‘Vikings’ conflates and blurs the distinction between eighth- and 12th-century pirates. Tenth-century kings based in Dublin and Christian rulers such as Cnut, all of whom lived in very different societies, had different belief systems and political and economic objectives. Each of these contexts needs to be dealt with on its own terms and not within a 19th-century construct that has more than a hint of racist essentialism to it.

It's also, insofar as I can infer from this article, religious exceptionalism.

3 Like • Comment

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