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What is the best layman interpretation of any mathematical concept you have seen?

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I had a student when I was a GTA for a vector calc course describe the average value of a multivariable function as the height of water in a fish tank once the waves have settled. I thought that was a super clear way of putting it.
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The metric axioms:

1. d(x,x)=0. You don't have to walk, to get to where you already are.
2. d(x,y)=d(y,x). You could walk over to your mate, but he could just as easily walk over to you.
3. d(a,c) <= d(a,b)+d(b,c).  You don't make your trip any shorter, by taking a detour.

Courtesy of my undergrad. topology lecturer.
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"you can't comb a hairy ball flat without creating a cowlick"

Condenses a bunch of vector analysis into something very approachable.
by
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Even if it’s not 100% accurate, I personally love putting-map-on-the-ground exercise as an example of Banach’s fixed-point theorem.
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Fundamental theorem of calculus: sum up all the little bits of change to get the total change.
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Cauchy-Schwarz: If you want to push a trolley along a track, it is best to push in the direction of the track.
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When my little brother yelled "Mommy, John ate one of my candies and I have only two left.", I knew I have successfully taught him subtraction.  It was very satisfying.
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Solving a differential equations is like solving a chess "mate in X" problem, except instead of figuring out future moves you're asked to figure out the past moves that lead you to this situation.
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You don’t do any calculations in manifolds. You first teleport to R^n do the calculation there and then teleport back with the corresponding answer.
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Triangular inequality: The sum of any two sides of a triangle is greater than the third side.

Explanation: If you are standing on one corner of a triangular field and you want to go to another corner, you'd rather walk straight towards that corner instead of going to the third corner and then to the destination corner. Because we know that going straight to the destination is the shortest path.

Alternate explanation: Suppose you have a string and the ends are tied taut to nails in a straight line. Now if you pinch anywhere on the string and pull it, stress will build in the string and will break the string. This is because the limited length of the string is not enough to cover the trace made by pulling the string which is basically a triangular trace.

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