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|Theorem: All positive integers are equal.Proof: Sufficient to show that for any two positive integers, A and B, A = B.Further, it is sufficient to show that for all N > 0, if A and B (positive integers) satisfy (MAX(A, B) = N) then A = B.Proceed by induction.If N = 1, then A and B, being positive integers, must both be 1. So A = B.Assume that the theorem is true for some value k. Take A and B with MAX(A, B) = k+1. Then MAX((A-1), (B-1)) = k. And hence (A-1) = (B-1). Consequently, A = B.

|Theorem : All numbers are equal to zero.Proof: Suppose that a=b. Thena = ba^2 = aba^2 - b^2 = ab - b^2(a + b)(a - b) = b(a - b)a + b = ba = 0Furthermore if a + b = b, and a = b, then b + b = b, and 2b = b, which mean that 2 = 1.

|Theorem: 3=4Proof:Suppose:a + b = cThis can also be written as:4a - 3a + 4b - 3b = 4c - 3cAfter reorganizing:4a + 4b - 4c = 3a + 3b - 3cTake the constants out of the brackets:4 * (a+b-c) = 3 * (a+b-c)Remove the same term left and right:4 = 3

|Theorem: 1$ = 1c.Proof:And another that gives you a sense of money disappearing.1$ = 100c= (10c)^2= (0.1$)^2= 0.01$= 1cHere $ means dollars and c means cents. This one is scary in that I have seen PhD’s in math who were unable to see what was wrong with this one. Actually I am crossposting this to sci.physics because I think that the latter makes a very nice introduction to the importance of keeping track of your dimensions.

|Theorem: 1$ = 10 centProof:We know that $1 = 100 centsDivide both sides by 100$ 1/100 = 100/100 cents=> $ 1/100 = 1 centTake square root both side=> squr($1/100) = squr (1 cent)=> $ 1/10 = 1 cent Multiply both side by 10=> $1 = 10 cent

|Theorem: n=n+1Proof:(n+1)^2 = n^2 + 2*n + 1Bring 2n+1 to the left:(n+1)^2 - (2n+1) = n^2Substract n(2n+1) from both sides and factoring, we have:(n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)Adding 1/4(2n+1)^2 to both sides yields:(n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2 = n^2 - n(2n+1) + 1/4(2n+1)^2This may be written:[ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2Taking the square roots of both sides:(n+1) - 1/2(2n+1) = n - 1/2(2n+1)Add 1/2(2n+1) to both sides:n+1 = n

|Theorem: 1 + 1 = 2Proof:n(2n - 2) = n(2n - 2)n(2n - 2) - n(2n - 2) = 0(n - n)(2n - 2) = 02n(n - n) - 2(n - n) = 02n - 2 = 02n = 2n + n = 2or setting n = 11 + 1 = 2

|Theorem: 4 = 5Proof:-20 = -2016 - 36 = 25 - 454^2 - 9*4 = 5^2 - 9*54^2 - 9*4 + 81/4 = 5^2 - 9*5 + 81/4(4 - 9/2)^2 = (5 - 9/2)^24 - 9/2 = 5 - 9/24 = 5

|Theorem: All numbers are equal.Proof: Choose arbitrary a and b, and let t = a + b. Thena + b = t(a + b)(a - b) = t(a - b)a^2 - b^2 = ta - tba^2 - ta = b^2 - tba^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4(a - t/2)^2 = (b - t/2)^2a - t/2 = b - t/2a = bSo all numbers are the same, and math is pointless.

|Theorem: 1 = -1Proof:1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = 1^ = -1Also one can disprove the axiom that things equal to the same thing are equal to each other.1 = sqrt(1)-1 = sqrt(1)Therefore 1 = -1As an alternative method for solving:Theorem: 1 = -1Proof:x=1x^2=xx^2-1=x-1(x+1)(x-1)=(x-1)(x+1)=(x-1)/(x-1)x+1=1x=00=1=> 0/0=1/1=1