A quesion of i

[googol]

Euler's notation, i, represents (-1)^.5, but what is i^2? 1 or -1. Does any body have a proof? I tried the following...

Given i = (-1)^.5 and a = i,

a = i

a^2 = i^2

a^2 = -1 (we assume i^2 = -1)

a = (-1)^.5

(-1)^.5 = (-1)^.5

...So is this proof?

I need help.

[HallsofIvy]

Euler's notation, i, represents (-1)^.5, but what is i^2? 1 or -1. Does any body have a proof? I tried the following...

Given i = (-1)^.5 and a = i,

a = i

a^2 = i^2

a^2 = -1 (we assume i^2 = -1)

a = (-1)^.5

(-1)^.5 = (-1)^.5

...So is this proof?

I need help.

The DEFINITION of a^(.5) is "the (positive) number whose square is a". If i is defined as (-1)^(.5), then, by that definition, i^2= -1. What you give appears to be going the wrong way: you start by assuming that i^2= -1 and the reduce to (-1)^.5= (-1)^.5.

That technique, sometimes called "synthetic proof", often used in proving trig identities, of starting from what you want to prove and then deriving an "obviously true statement (-1)^.5= (-1)^.5, only works if every step is "reversible". Here that is not the case. a^2= -1 could be reduced to a= i or a= -i.