246A, Notes 0: the complex numbers

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Kronecker famously wrote, “God created the natural numbers; all else is the work of man”. The truth of this statement (literal or otherwise) is debatable; but one can certainly view the other standard number systems $latex {{bf Z}, {bf Q}, {bf R}, {bf C}}&fg=000000$ as (iterated) completions of the natural numbers $latex {{bf N}}&fg=000000$ in various senses. For instance:

• The integers $latex {{bf Z}}&fg=000000$ are the additive completion of the natural numbers $latex {{bf N}}&fg=000000$ (the minimal additive group that contains a copy of $latex {{bf N}}&fg=000000$).
• The rationals $latex {{bf Q}}&fg=000000$ are the multiplicative completion of the integers $latex {{bf Z}}&fg=000000$ (the minimal field that contains a copy of $latex {{bf Z}}&fg=000000$).
• The reals $latex {{bf R}}&fg=000000$ are the metric completion of the rationals $latex {{bf Q}}&fg=000000$ (the minimal complete metric space that contains a copy of $latex {{bf Q}}&fg=000000$).
• The complex numbers $latex {{bf C}}&fg=000000$ are the algebraic…

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IBM’s Big Data University: Free Online Learning With Over 400,000 Students – Forbes

It is widely forecasted that a shortage of skills in data science and analytics will mean a great deal of money is wasted through missed opportunities in coming years. Traditional academic establishments have begun to move to fill the gap. However, most courses teaching the hot topic skillsets such as […]

Sourced through Scoop.it from: www.forbes.com

Great free content here:

Why there is no Hitchhiker’s Guide to Mathematics for Programmers

Math ∩ Programming

For those who aren’t regular readers: as a followup to this post, there are four posts detailing the basic four methods of proof, with intentions to detail some more advanced proof techniques in the future. You can find them on this blog’s primers page.

Do you really want to get better at mathematics?

Remember when you first learned how to program? I do. I spent two years experimenting with Java programs on my own in high school. Those two years collectively contain the worst and most embarrassing code I have ever written. My programs absolutely reeked of programming no-nos. Hundred-line functions and even thousand-line classes, magic numbers, unreachable blocks of code, ridiculous code comments, a complete disregard for sensible object orientation, negligence of nearly all logic, and type-coercion that would make your skin crawl. I committed every naive mistake in the book, and for all my obvious…

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Learning Programming — Finger-Painting and Killing Zombies

Math ∩ Programming

Zmob, my first (and only) original game.

By the end, the breadth and depth of our collective knowledge was far beyond what anyone could expect from any high school course in any subject. 

Education Versus Exploration

I’m a lab TA for an introductory Python programming course this semester, and it’s been…depressing. I remember my early days of programming, when the possibilities seemed endless and adding new features to my programs was exciting and gratifying, and I brimmed with pride at every detail, and I boasted to my friends of the amazing things I did, and I felt powerful. The world was literally at my fingertips. I could give substance to any idea I cared to entertain and any facet of life I wanted to explore. I had developed an insatiable thirst for programming that has lasted to this very day.

My younger self, if programming were more noodley.

The ironic thing is…

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Overview of GE’s Change Acceleration Process (CAP)

Bob Von Der Linn's HPT Blog

In 1989-90, under the direction of Jack Welch, GE launched “Work-Out” – a team based problem-solving and employee empowerment program modeled after the Japanese quality circles model that was in vogue at the time.  Work-Out was a huge success and Welch was frustrated by the rate of adoption through the business.  Welch, the visionary, realized that GE (and everyone else!) was entering an era of constant change, and that those who adapted to change the fasted would be the survivors.  He commissioned a team of consultants (including Steve Kerr, who was to become GE’s first Chief Learning Officer) to scour industry and academia to study the best practices in change management and come back to GE with a tool kit that Welch’s managers could easily implement.  The result was the Change Acceleration Process, commonly referred to within GE simply as “CAP.”[1]

The Change Effectiveness Equation

The team studied hundreds…

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Logarithms Explained, and the Associative Property of Multiplication

The Trickle-Down

So much advanced math relies on a firm grasp of basic Algebra and Algebra II.

Today, lets take a look at logarithms!

So what are logarithms? Well, first let’s look at exponential equations, such as $latex 2^x = y$ where the 2 is a base. We all know that for example, $latex 2^3 = 8$. A general form is $latex b^x = y$ where b is the base. Well, with logarithms, the format is $latex log_b y = x$. So for $latex 2^3 = 8$, we would express that with logarithms as $latex log_2 8=3$. Fun, isn’t it! The logarithm is the number that the base is raised to a power by to equal a given number; in the example above, the base 2 is raised by the power 3 to equal the number 8.

So the tricky part is that you get rules like $latex log_b y + log_b…

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Jeff Bezos on Leading for the Long-Term at Amazon

Wish Bezos did more of these…