Thursday, March 30, 2006 1:33:19 PM
But does 1 = 2 ?
1st Proof
x
=
1
set x equal to 1
x2
=
x
multiply both sides by x
x2 - 1
=
x - 1
subtract 1 from both sides
(x - 1)(x + 1)
=
x - 1
separate left side into factors
x + 1
=
1
divide both sides by (x - 1)
1 + 1
=
1
substitute 1 for x
2
=
1
2nd Proof
(-1)(-1) = 1 the square of -1 is 1
-1 = 1/-1 divide both sides by -1
-1/1 = 1/-1 identity operation; for all real (or complex) x, x = x/1
i/1 = 1/i take the square root of both sides (i=sqrt(-1))
i/2 = 1/2i divide both sides by two
i2/2 = i/2i multiply both sides by i
-1/2 = 1/2 substitute -1 for i2 and 1 for i/i
-1/2 + 3/2 = 1/2 + 3/2 add 1 1/2 (3/2) to both sides
1 = 2
3rd Proof
x2 = x+x+x+...+x (x times) definition of x2; x not equal to zero
2x = 1+1+1+...+1 (x times) take derivative of both sides;
derivative of xn = nxn-1
2x = x x = 1+1+1+...+1 (x times)
2 = 1 divide both sides by x (x not equal to zero)
I'm Done :)
1st Proof
x
=
1
set x equal to 1
x2
=
x
multiply both sides by x
x2 - 1
=
x - 1
subtract 1 from both sides
(x - 1)(x + 1)
=
x - 1
separate left side into factors
x + 1
=
1
divide both sides by (x - 1)
1 + 1
=
1
substitute 1 for x
2
=
1
2nd Proof
(-1)(-1) = 1 the square of -1 is 1
-1 = 1/-1 divide both sides by -1
-1/1 = 1/-1 identity operation; for all real (or complex) x, x = x/1
i/1 = 1/i take the square root of both sides (i=sqrt(-1))
i/2 = 1/2i divide both sides by two
i2/2 = i/2i multiply both sides by i
-1/2 = 1/2 substitute -1 for i2 and 1 for i/i
-1/2 + 3/2 = 1/2 + 3/2 add 1 1/2 (3/2) to both sides
1 = 2
3rd Proof
x2 = x+x+x+...+x (x times) definition of x2; x not equal to zero
2x = 1+1+1+...+1 (x times) take derivative of both sides;
derivative of xn = nxn-1
2x = x x = 1+1+1+...+1 (x times)
2 = 1 divide both sides by x (x not equal to zero)
I'm Done :)
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