Proof that -1 = 1 and 1 = 3

I know something is wrong in the following proofs.Can't figure out which step is wrong.

Proof of -1 =1:

Proof of 1=3:

7 Answers

383
Soumyabrata Mondal ·

for 2nd one (2) step

2305
Shaswata Roy ·

@Soumyabrata Mondal
Even I had thought of that before.
But you see,
\dpi{200} \fn_cs e^{i\pi} = -1\Rightarrow log(-1) = i\pi
Hence log(-1) does have a value.

229
Dwijaraj Paul Chowdhury ·

For 1st one.....
The first question was given in ISI
you cant express two negative nos which are in square root
form as two different roots...i mean√(-1*-1) cannot be written as root-1 *root -1...nd @Roy the question is --"Proof that"?? :-|

36
Karan Bhuwalka ·

√ (-1).(-1) equals 1

1357
Manish Shankar ·

a/b=√a/√b for positive values of a and b only

229
Dwijaraj Paul Chowdhury ·

for the 1st proof
Dekh log er value replace korle
3i*pi=1i*pi
Now LHS=(0,3*pi)
And RHS=(0,pi) by multiplication of complex nos. But as they don't represent the same complex nos. on an argand plane, they cannot be equal. :. Contradiction. LHS is not equal to RHS.

  • Shaswata Roy Thanks for stating the obvious :P....It's known that LHS is not equal 2 RHS . I am asking why they are not equal (i.e. what mistake had I made while taking the log).As I had said b4 2 Soumyabrata, it has got something 2 do with one to one correspondence property of log or something like that(don't know).
337
Sayan Sinha ·

I think, for the 2nd one,

log is not possible for a negative number.

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