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.

January 8, 2024 at 9:58am

#1062064

Math Hole

I just thought this one would be appropriate for this blog.

How to beat maths anxiety
Getting stressed about maths is so common, yet there’s no need. With the right approach, you can even start enjoying it

According to the author's bio, she lives in New York (or at least did in October of last year), but the article uses Brit spellings such as "maths." Just roll with it.

It's a fairly long article, but there's very little actual math in it. The first section's purpose is to explain what this anxiety is:

Do this maths problem: 34 minus 19. Do it in your head without paper, and then imagine another person watching you. How does that make you feel? Mark Ashcraft, a US psychologist who studied maths anxiety, described how people in his studies responded to solving similar problems. Many showed ‘unease or apprehension’. They had ‘trembling hands, nervous laughter, and so forth,’ he wrote.

I am in no way frightened of math. More advanced topics are beyond me, sure, but if I encounter "what's 34 minus 19" or "what's 20% of 130," or whatever, I just do it. Unless, of course, someone is watching me, waiting to see if I'm slow or will get it wrong, so they can feel superior. So yeah, I kinda get that.

You may not have realised that maths anxiety had a name at all, but if you’ve tended to go out of your way to avoid maths, it’s likely affected your life.

As it's a part of nearly everything, it surely has.

A while back, so the story goes, to compete with the quarter-pounder burger at you know where, a smaller fast food chain started selling a 1/3-pound burger (both measures were weight before cooking, but that's not relevant to the story; presumably, each would lose the same percentage of mass on the grill). They set it for the same price, figuring people would switch over to pay the same amount for more meat. But the promotion flopped, and it flopped because a lot of people saw 1/4, and they saw 1/3, and they concluded that since 4 is greater than 3, why should they pay the same thing for less burger?

For a long time, I didn't believe this story. Obviously, some people are that ignorant, but surely not a majority. Most people know that 1/3 is greater than 1/4, right? It's right there on your measuring cups, for one thing. The failure must have had another reason: maybe the 1/3 pound burger tasted like shit. Maybe it didn't, but people wanted to stick with the familiar. Maybe they were fine with a 1/4-pound patty, but didn't want the extra calories. Maybe they didn't trust the rival (it was A&W, incidentally) to have the staying power of McDonald's.

But no.

People really are that ignorant. Maybe that's because of math anxiety; I don't know. I do know that when I had this confirmed, I still didn't want to believe it. It's not that I think people are unbearably stupid, or that I'm somehow superior because I can subtract 19 from 34 really quickly. Ignorance isn't the same thing as stupidity. But stubborn refusal to fix ignorance? There, I start having issues.

Today, about 93 per cent of US adults say they have some amount of maths anxiety, and 17 per cent report high levels.

That seemed high even to me, so I checked the source. Indeed, the study it came from said 93%. What's worse, though, is that the same report goes on to say, "In a sample of adolescent apprentices in the United Kingdom, approximately 30% of the study participants reported high math anxiety, and a further 18% were at least somewhat affected by it." While the sample is probably different, if you do the math (yes, really), that implies 48% in the UK. Way lower.

So, anyway, after the explanation, almost as long as the thesis you have to write on a recipe page, there's the "what to do" section. I won't quote extensively from that, either, but I do have some comments:

Learning a foreign language or reading Beowulf can be just as cognitively demanding or complex as manipulating numbers, yet people tend not to be as intimidated by these sorts of linguistic activities.

I was way more intimidated at the thought of learning a foreign language, in school, than I was about math. Mostly, this was because of the group nature of the activity, I think.

Long enmeshed in the myth is the notion that it’s not just that some people are better at maths, but that those people fall into certain groups. Most common is the cultural belief that men thrive more in maths (along with other STEM fields) than women.

The math champion of my high school was a girl. She was cute, too. Sometimes I wonder what became of her. I know this is anecdata, but sometimes all you need is one example to break a stereotype. In this case, the stereotype is "girls are worse at math." Point is, I knew from personal experience that this wasn't true.

Expressive writing is when you write for a specific period of time about your emotions and thoughts that come up in response to a stressful maths situation, such as sadness that you’re not good or smart enough, stress that you’re going to fail, or anxiety or anger that it’s too hard.

See, even in an article about math, we can slip a little writing in.

Teach maths to someone else

I find this to be good general advice: if you want what you're learning to really sink in, explain it to someone else.

So, like I said, lots more at the link. Even if you have math anxiety and aren't ready or willing to tackle it, there's good information.

No pressure. There won't be a test later. Not here, anyway. Just be wary of fast food joints trying to trick you.

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