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

Aug 12, 2020

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

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

)

@QiaochuYuan

Aug 12, 2020

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

philemon

@phil49472744

Aug 12, 2020

can you explain this one?

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

)

@QiaochuYuan

Aug 12, 2020

two consistency checks: this makes the laplace expansion continue to work for 1x1 matrices, and it's consistent with the determinant being "block-multiplicative" in that the determinant of a block-diagonal matrix should be the product of the determinants of the blocks

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

Replying to

and

depending on how you define the determinant this follows from some of the other results. the determinant of the empty matrix is a sum over permutations of the empty set, of which there's one, and the unique term of this sum is the empty product, which is 1

10:56 PM · Aug 12, 2020Twitter Web App

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

Replying to

and

if you define the determinant in terms of exterior powers, the zeroth exterior power of the zero-dimensional vector space is the one-dimensional vector space (this is another "empty product" phenomenon) and it sends the identity to the identity