Conversation

Aug 12, 2020

thread: fun facts about zero / the empty set. i will die on all of these hills 1. 0^0 = 0! = {0 choose 0} = 1, these are all variations of the same fact which is that there's a unique function from the empty set to itself, which does nothing. this function is not constant

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Eliezer Yudkowsky

@ESYudkowsky

· Aug 12, 2020

How many different ways are there to choose 0 items from an empty set?

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166

QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

2. the sum of zero terms (the empty sum) is 0; the product of zero terms (the empty product) is 1

10:03 PM · Aug 12, 2020Twitter Web App

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QC (in Berlin 7/29-??? w/ COVID

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3. the degree of the zero polynomial is -∞

QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

Aug 12, 2020

4. the maximum of zero numbers is -∞; the minimum of zero numbers is +∞

QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

Aug 12, 2020

5. the determinant of the 0x0 matrix (the empty matrix, with no entries) is 1

QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

Aug 12, 2020

6. the empty graph is disconnected, and so is the empty topological space, for the same reason that 1 isn't a prime number

QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

Aug 12, 2020

7. there's a different empty manifold in each dimension: a zero-dimensional empty manifold, a 1-dimensional empty manifold, etc. (and of course none of them are connected or path-connected)

QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

Aug 12, 2020

8. the zero-dimensional vector space (consisting only of the vector 0) has a unique basis, which is empty

QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

Aug 12, 2020

9. (this one is new to me, thanks

@achambertloir

!) the zero endomorphism of the zero banach space has operator norm 0; the endomorphisms of the zero banach space do still form a unital banach algebra but only if you relax the unit axiom to |id| \le 1

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Antoine Chambert-Loir

@achambertloir

· Aug 12, 2020

Replying to @achambertloir and @QiaochuYuan

By the way, for the operator norm of an endomorphism, the supremum is taken in R_+ adjoined +infinity, so that the norm of the identity endo of the null space is 0.

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omg flashback to teaching pre-algebra and finally figuring out how to explain why x^0 = 1 (they always expected it to be 0)... remind them that x = x*1, so x^3= x*x*x*1 x^2 = x*x*1 x^1 = x*1 x^0 = 1 for some reason this was by far the most convincing argument