A few properties of the natural log, ln(x), are as follows
Now consider the following proof.
All numbers are equal.
proof
1 = (-1)^2 = 1^2
Therefore, ln( (-1)^2 ) = ln( 1^2 ). Using log properties, 2 ln( -1 ) = 2 ln(1), or ln(-1)=ln(1).
Now remembering that ln is a one-to-one function we have -1=1.
0 = 1 + (-1) = 1 + 1 = 2, i.e., 0=2. Also 1 = 2/2 = 0/2 = 0. Thus 0=1.
All integers can be constructed from 1 and -1 (n=1+1+...+1 n times); therefore, because 1=0, n=0 for all integers n. Because -1=1, the same holds for the negative integers. Thus all integers are equal.
Further, all rationals are ratios of integers (a/b) by definition. Now because a=0 and b=1 for all integers a and b, a/b=0/1=0. Thus all rationals are equal.
Next, all reals are the limits of sequences of rationals. Because all rationals are equal to 0, all sequences are constant sequences converging to 0. Thus all reals are equal.
Finally, all complex numbers are of the form a+bi where a,b are reals. Thus all complex numbers are of the form 0+0i=0, and all complex numbers are equal.
Q.E.D.
Can you find a flaw in this proof? If not, you may have discovered that the U.S. national debt is $0.
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