More intuitive explanations!

6th Jan 2012

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11 comments, sorted by Click to highlight new comments since: Today at 10:30 AM

Steven Strogatz did a series of blog posts at NY Times going through a variety of math concepts from elementary school to higher levels. (They are presented in descending date order, so you may want to start at the end of page 2 and work your way backwards.) Much of the information will be old hat to LWers, but it is often presented in novel ways (to me, at least).

Specifically related to this post, the visual proof of the Pythagorean theorem appears in the post Square Dancing.

FWIW, there's a nice proof of Bayes' theorem in Russel and Norvig's textbook, which I haven't seen posted here yet.

Spencer Greenberg of Rebellion Research:

The behavior of [a] machine is going to depend a great deal on which values or preferences you give it. If you try to give it naive ones, it can get you into trouble.

If I say to you, "Would you please get me some spaghetti?", you know there are other things in the world that I value besides spaghetti. You know I'm not saying that you should be willing to shred the world to get spaghetti. But if you were to code naively into an extremely intelligence machine as it's

onlydesire, "Get me spaghetti," it would stop at absolutely nothing to do that.So the danger is not so much the danger of a machine being evil or Terminator. The danger is that we give it a bad set of preferences... that lead to unintended consequences. Or [maybe] it's built from the ground up with preferences that don't reflect the preferences of most of humanity — maybe only the preferences that only a small group of people cares about.

5) Proof of Euler's formula using power series expansions.

I forgot that this is only insightful if you already realized the following:

- (3+4i) =
- (r, angel) =
- (sqrt((3^2)+(4^2)), arctan(4/3)) =
- (5, 53.1301) =
- r(cos(x)+isin(x)) =
- 5(cos(arctan(4/3))+isin(arctan(4/3))) =
- 5e^(arctan(4/3)*i) =
- e^ln(5)*e^(arctan(4/3)i) =
- e^(ln(5)+arctan(4/3)i)

Only the cartesian, polar and cos-sin forms were obvious to me, and I was still able to make sense of the Taylor series proof.

The post on two easy to grasp explanations on Gödel's theorem and the Banach-Tarski paradox made me think of other explanations that I found easy or insightful and that I could share them as well.

1) Here is a nice

proof of the Pythagorean theorem:2) An easy and concise

explanation of expected utility calculationsby Luke Muehlhauser:3)

Micro- and macroevolutionvisualized.4)

Slopes of Perpendicular Lines.5)

Proof of Euler's formulausing power series expansions.6)

Proof of the Chain Rule.7)

Multiplying Negatives Makes A Positive.8)

Completing the SquareandDerivation of Quadratic Formula.9)

Quadratic factorization.10)

Remainder Theorem and Factor Theorem.11)

Combinations with repetitions.12)

Löb's theorem.