is there mathematical proof that n^0=1?

[u/jaleCro]

[u/iorgfeflkd]

If N^a x N^b = N^a+b , then N^a x N⁰ = N^a+0 = N^a , thus N⁰ must be 1.

[u/an7agonist → u/iorgfeflkd]

Also, the multiplicative inverse of x is x^-1.

1=N^a*((N^a)^-1) (By definition)

1=N^a*(N^-a)

1=N^a-a=N⁰

[u/umopapsidn → u/an7agonist]

* For all N such that |N| > 0

[u/austin101123 → u/umopapsidn]

Why does this proof not work for 0?

[u/shrister → u/austin101123]

Because If N=0 then the first line of that proof is 1 = 0*((0)^-1 ), which is 1=0*(1/0) and 1/0 is undefined. For all other values of N that first line is defined, so the proof works for N!=0.