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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 6 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 8 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.

August 26, 2020 at 12:02am

#991593

For the Rest of It

You might have thought yesterday's article about quantum physics put you to sleep. Well, wait'll you see today's...

https://www.domino.com/content/benefits-of-napping/

What Happens to Your Body When You Take Naps Every Single Day?

It sleeps. Duhhh.

As often as we talk about the benefits of sleep, more than a third of Americans are not getting the proper amount of shut-eye.

That's because they're working three jobs just to live.

If the thought of adding one more thing to your already busy schedule is making you stress out, you can consider naps as a natural way to recharge for the day.

Maybe during your commute?

"Like meditation, it can be [used] as a quiet time in the middle of a chaotic day."

What a coincidence! Every time I've tried to meditate I've fallen asleep.

However, taking a midday nap doesn’t mean you can sleep the day away.

Do people actually do this? I'm retired and don't have a set schedule most days, and I still can't sleep more than 9 hours, max. Usually 7-8. In two shifts.

While you may want to sneak a small nap into your day, you might have a hard time getting a little shut-eye if it’s not something your body is used to doing.

My body was used to me fighting to stay awake through a slump every day between 4-5 pm or so, and then being forced to wake up before 8 am. It never worked well for me, as much as I tried to conform.

Basically, adding a nap into your daily routine will give your body a major health boost. After six months, the long-term benefits of napping kick in. Breese notes one study on Greek adults that found a short nap during the day reduces the risks of dying from heart disease and regularly getting more rest may increase your sex drive.

Well, that's useless to me.

Incidentally, can anyone tell me why they people call it "beauty sleep?" This has never made sense to me. I get the feeling it's used sarcastically, or perhaps apologetically. "Oh, just like every other human being on the planet, I need to sleep, but admitting that might make me seem weak, so I have to pretend there's a purpose to it." Or something like that. Like I said, I never understood it, and it's a phrase that's been around since at least my childhood.

Well, I'll sleep later. First, drinking. Which doesn't actually help one sleep, but it's worth it for its own sake.

3 Like • Comment • View 7 comments

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