Complex Numbers (Book)

Blog Calendar

◄ April ► 2020
S	M	T	W	T	F	S
			1	2	3	4
5	6	7	8	9	10	11
12	13	14	15	16	17	18
19	20	21	22	23	24	25
26	27	28	29	30
Archive				RSS

About This Author

Contact Me

Come closer.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 14 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 16 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.

April 18, 2020 at 12:03am

#981372

Historical Badass

One of the most annoying things about the history of mathematics and science is that women's work in those fields is largely discounted or forgotten.

https://massivesci.com/articles/gabrielle-emilie-du-chatelet-voltaire-newton-phy...

Meet Émilie du Châtelet, the French socialite who helped lay the foundations of modern physics
She improved on Newton's work and outclassed Voltaire

One of the few exceptions is Marie Curie, who of course was also French. I don't think I'd ever heard of Émilie du Châtelet, so I found this article interesting.

Mathematician, physicist, French aristocrat, manager of household, wife, mother, and philosopher, du Châtelet lived her short life to its fullest.

Had she lived just a little later, being a French aristocrat, her life might have been even shorter thanks to La Révolution. So would she.

It is clear, though, that she was interested in many “unfeminine” subjects. Home-educated, she learned multiple languages, including German, Greek, Latin, and Italian...

These days, home-schooling is associated with learning less, not more.

She was dedicated to her studies, and creative in finding resources – she used her mathematical talents to succeed in gambling and used her profits to buy textbooks and lab supplies.

I think I would have liked her.

Her time in Paris also brought her in contact with her most famous lover – the philosopher Voltaire.

Describing Voltaire as a "philosopher" is probably a slight to actual philosophers. "Rabble-rouser" might be more accurate. Oh, I know, Enlightenment values, blah, blah, but the guy seems like he was a bit of a tool, even if he was sometimes right.

When he had to flee Paris for his writing on England, which the French royalty and church took as a threat, he went to du Châtelet’s chateau in Cirey-sur-Blaise. Her husband apparently didn’t mind; Voltaire and du Châtelet were together for the next 15 years, all the while building a library and running scientific experiments.

I'm including this here in case you had any doubt in your mind that this story is about French people.

After the success of her previous publications, du Châtelet found herself looking for a new challenge. She decided to take on Newton’s Principia and its accompanying commentaries.

The article explains that Newton's work was in Latin, and she made it accessible to French readers.

Writing sporadically over four years, du Châtelet added in detailed proofs of ideas Newton mentioned offhand and integrates the latest works of her contemporaries, which built on and supported Newton’s ideas.

Now I'm wondering who might have done that for English readers. As it is, I'm beginning to get some idea of why France was scientifically ahead of England for many years, despite Newton being English.

Yet, it would be ten years after its completion before the public would see her masterpiece. She had finished her writing in a frenzy, working until the wee hours of the morning, while pregnant with her fifth child, by Jean François de Saint-Lambert (she and Voltaire had separated). She was 42 – and at the time, a pregnancy at that age was a death sentence.

At the time, any pregnancy was risky. This is one reason why reliable birth control should be considered one of the greatest inventions of modern times.

And in case the science thing wasn't enough, she was also apparently an early advocate of feminism, as it might have been understood then:

She advocates for women’s education: “This education would render enormous service to the entire human race. Women would be enriched by it and men would find new respect for them. Male–female interaction, which too often polishes manners while weakening and shrinking minds, would then rather serve to expand their knowledge.”

It took a while for that to happen. Equality is a slippery thing, sometimes, and is viewed through the lenses of a period's culture and times -- hence, you have people like Voltaire advocating for the common people while insulting them, or Jefferson writing high-minded polemics about freedom while owning slaves. I'm sure one day our quaint notions of what equality looks like will be scoffed at by future generations (if we have future generations), but hopefully some people will see the incremental progress we've made for what it is.

4 Like • Comment • View 2 comments

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.