Proofs by mathematical induction.

We describe the principal of mathematical induction and give several examples.

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[piotryjak]

It would be so great if you could make it easier to find previous parts of your courses. A playlist, or even an old-fashioned link in the description to a first video of you multi part content would be so helpful. Great content, love how fluent you are, just freaking badass, you mister are a rock star.

[buxeessingh]

I used to use that thumbnail in class. Very few admitted they got the joke.

I want to emphasise the importance of remembering the base case in the induction step. Often, how you would perform your induction can be discovered from using the n = 1 case in a proof for n = 2. That is where you would use the base case in the induction step.

[xcheese]

A man deft in liquor production
Runs stills of flawless construction.
The alcohol boils
Through magnetic coils.
He says that it’s “proof by induction.”

[eleazaralmazan]

By far my favorite proof technique! #MathematicalInducation

[yakovify]

I absolutely love proofs by induction and especially their variants such as the ones for graph theory, trees and ordinals.

But my favorite has to be what I like to call Analytic Induction.
It goes as follows:
Let X be a connected topological space and P(x) is some property of every point x in X.
Assume that there exists at least one element y in X such that P(y) is true. (base case).
Also assume that if P(x) is true for some x then there exists an open neighborhood U of x in X such that P(z) is true for all z in U.
Finally assume that if P(x_n) is true for some sequence of x_n's then P(z) is true for all limit points z of x_n.

If you can show the above you have successfully proven that P(x) is true for all x in X.

I love this because it gives me such a vivid image of what I am proving, the property P(x) spreading from point to point till it covers all of X. Absolutely beautiful.

[tomatrix]

I just noticed the thumbnail is a reference to electrical induction

[MrRyanroberson]

Interestingly, the angle sum rule doesn't require that the figure be convex, so long as it is euclidean (which is a bit of a circular definition, since euclidean space can be defined as that which obeys the angle sum for all polygons)

[rockinroggenrola]

If I'm not mistaken, didn't you make another video about induction before?

[martinnyberg]

@MichaelPenn Please tell us more about all your pretty chalk! It looks very nostalgic, and soft and comfortable. I haven’t taught using chalk since the 1990s; all I get to use these days are stinkin’ whyteboard pens. 😁

[lionking]

Try India's exam 'JEE ADVANCED' maths problems... U will find very good calculus problems out there!!

[judysalazar]

Professor Penn, in the induction hypothesis we assume that there exists some natural number k such that p(k) implies p(k+1)?

[Hobbit]

I like to watch these videos even if i understand very little ;) its meditating

[AntoshaPushkin]

Isn't strong induction just a regular induction, but instead of P(n) we make a new predicate Q(n) which is
Q(n) = for all k, k ≤ n => P(k)
prove base and step for Q, and then we get Q(n) implies P(n) for all n?

[michalbotor]

(3:55) all horses are the same color (all people are the same height)

[tomasbeltran]

I left at 15:30. I'm writing down the examples.

[nathanisbored]

I like to imagine induction as like an infinite row of dominoes. For it to work, the dominoes need to be evenly spaced (which is why n and n+1 should be integers). Proving the induction step is like setting up the dominoes so that they are aligned, and showing the base case is like knocking over the first domino (though you can really do these in either order).

[poi_aithhkunnnRVC]

23:34 broken "good place to stop" first

[gaborendredi]

In high school my math teacher once made an induction proof like this. He proved for n=2 as base case. Then he showed that P(n) => P(n^2) and P(n) => P(n-1). He claimed, that this way was still more easy than to show the usual P(n) => P(n+1). Unfortunately I don’t remember the statement he proved this way.

[tylershepard]

Hey now be careful when throwing shade at us Electrical Engineers. You are on the internet after all...

[aamierulharith]

"One is even" press x to doubt