Sort of like, you know, today we were to wake up and discover that there were only nine commandments, and the 10th was planted there by the devil. And that is a false proof. So they covered it up, and they denied the result. Well bigger than 2 is bigger than 2, but, yeah, maybe that should have been a 2 minus.

If I put them together, I get a 9 by 10 rectangle. Are the triangles, like, drawn to scale?

Now the proofs that we covered last week in recitation, the problems set, were all examples of what are called direct proofs. Right away you should be suspicious.

This is the legend. All right, b is even, good. So they said there were no irrational numbers. And then you use that hypothesis, namely the p is false, to derive a falsehood. All right if a is even, what do I know about a squared? Anything that is reasonable, is fine by us.

Because of the conservation of area axiom? Would that change anything? So this is one of the big problems of proofs by picture. Now it sounds weird today, but back then, in ancient Greece, math was a religion, all right? But if you took a triangle, and you have side lengths one, the length of the hypotenuse was square root 2.

Lecture 2 Just, you know, be reasonable. B squared is even, then a is even. First it meant that their axioms were inconsistent. I guarantee you, if I measured on my screen, that 2 plus is bigger than the length of the tube.

But then, according to legend there was a Deep Throat. I got a contradiction here, somewhere? And that is a contradiction. And this is a little bit different. So that means that four divides a squared.

Then you just take steps for logical deductions forward until you arrive at a contradiction, something where you prove false equals true. Square root of 2 is not rational, so it must be irrational. You start with some axioms, you have some theorems you knew before or you proved along the way, and you make logical deductions until you get to where you want to go, the theorem.

So most of you have seen a proof of that, good. They liked rational numbers, you know, like one seventh?Lecture Video. The backscattering experiment of Rutherford is recreated in the classroom setting. Ping pong balls are used to represent alpha particles and Styrofoam balls connected to a series of strings represent nuclei in a piece of gold foil.

That equals, well, I got n squared plus n, here, over 2, plus 2 n plus 2.

And that equals n plus one times n plus 2 over 2, which is what we're trying to show. So we've completed, now, the inductive step. Lecture 2 I/O Model The I/O (Industrial Organization) Model adopts an external perspective to explain that forces outside of the organization represent the dominate influences on a firm's strategic actions and is based on the following four assumptions.

Hillsdale College was founded in by men and women who proclaimed themselves “grateful to God for the inestimable blessings resulting from the prevalence of civil and religious liberty and intelligent piety in the land,” and who believed that “the diffusion of sound learning is.

Lecture 2 • Short-Channel MOSFET Electrostatics – MOSFET Performance Metrics – Short-Channel Effects Reading: Taur & Ning, “Fundamentals of Modern VLSI Devices,” Cambridge Univ. Press, - multiple research articles (reference list at the end of this lecture).

Lecture 2 Electronic Devices and Circuits - S 17 Summary Why are IC’s made out of Silicon? • Two types of “carriers” (mobile charge particles): – electrons and holes • Carrier concentrations can be controlled over many orders of magnitude by addition “dopants”.

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