Page 198 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

We can prove this by contradiction. Let’s assume there is a real number z = y , and

z
e = x.

Since z = y , by possibly changing their names, we can assume y> z . Since x> 0,

e y x
= =1,
e z x

so by the rules for exponents

e y
= e y−z =1.
e z

By the definition of e ,

∞ ∞
1 1
e = =2 + > 2.
n! n!
n=0 n=2
Since y> z , y − z> 0, so

1= e y−z > 2 y−z > 1.

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