Strange Math

Equation 1.
a = b
a2 = ab
a2 - b2 = ab - b2
(a - b)(a + b) = b(a - b)
a + b = b
2 = 1

Equation 2.
sqrt(-1) = sqrt(-1)
sqrt(-1/1) = sqrt(1/-1)
sqrt(-1) / sqrt(1) = sqrt(1) / sqrt(-1)
sqrt(-1)*sqrt(-1) = sqrt(1)*sqrt(1)
-1 = 1

Equation 3.
0 = 0 + 0 + 0 + 0 ...
0 = (1-1)+(1-1)+(1-1)...
0 = 1 + (-1+1) + (-1+1) + (-1+1)...
0 = 1 + 0 + 0 + 0 + 0...
0 = 1

Equation 4.
Int( 1/x ) // Int = Indefinite integral
= Int( 1 * 1/x )
= x * 1/x - Int( x * ( -1/(x^2) ) ) // Integration by parts
= 1 + Int( 1/x ),
so that 0 = 1.

Equation 5.
1 = 1
1 = 1 + 0 + 0 + 0 ...
1 = 0 + 1 + 0 + 0 ...
1 = 0 + 0 + 1 + 0 ...
1 = 0 + 0 + 0 + 1 ...
1 = 0 + 0 + 0 + 0 ...
1 = 0

Equation 6a.
1 = 3 * 1/3  = 3 * .33333... = .99999....

Equation 6b.
   1.00000.....
-  0.99999.....
---------------------
   0.00000.....

Equation 6c.
N = 0.99999...
10N = 9.9999....
9N = 9      (all decimal portion is cancelled out)
N = 1

Equation 6d.
1) x = 3.99999...
2) 10 x = 39.99999... multiply both sides by 10.
3) 9 x = 36 subtract 1) from 2).
4) x = 4 divide both sides by 9.
5) 3.99999... = 4 substitute the value of x as given in 1)

Equation 7.
x*x = x + x + x + ... + x (x times)
d/dx (x*x) = d/dx ( x + x + x + ... + x )
d/dx (x*x) = d/dx (x) + d/dx (x) + ... d/dx (x)
2x = 1 + 1 + ... + 1
2x = x
2 = 1

Equation 8.
A = B <> 0
A^2 = A*B
A^2 - B^2 = A*B - B^2
(A+B)*(A-B) = B*(A-B)
A+B = B
B + B = B
2*B = 1*B
2 = 1

Equation 9.
x^2 - x^2 = x^2 - x^2
x(x-x) = (x+x)(x-x)
x = x + x
x = 2x
1 = 2

Equation 10.
Assume: x = y = 1;
x = y;
x^2 = xy; 
x^2 - y^2 = xy = y^2
Factor the terms:
(x+y)(x-y) =(y)(x-y)
Factor out the duplicate terms:
(x+y) = (y)
1 + 1 = 1;
Therefore: 2 = 1!

Equation 11.
a/b + c/d = (ad + bc)/bd,
1 = 0 + 1 = 0/0 + 1/1 = (1*0 + 0*1)/(0*1) = 0/0 = 0
Therefore: 1 = 0

Equation 12.
a*a - a*a = a*a - a*a
(a+a)(a-a) = a(a-a)
a+a = a

Equation 13.
a=b
Next, multiply both sides by a:
a^2=ab
Now, subtract b^2 from both sides:
a^2-b^2=ab-b^2
Factor the equations:
(a+b)(a-b)=b(a-b)
Divide both sides by (a-b):
a+b=b
Since a=b, replace all of the b's with a's:
a+a=a
Simplify:
2a=a
Divide by a:
2=1

Equation 14.
Consider the equation:
Ax^2 + Bx + C = 0, with AC != 0
Ax^2 + Bx + C = 0 ---> x = (-B +- V(BB-4AC))/2A
divide by x^2
C (1/x)^2 + B (1/x) + A = 0 ---> 1/x = (-B +- V(BB-4AC))/2C
Dividing those two formulas gives:

x (-B +- V(BB-4AC)) / 2 A
-------- = --------------------------
1/x (-B +- V(BB-4AC)) / 2 C

cancelling
x^2 = C / A
x = +- V(C/A)

Equation 15.
16 - 36 = 25 - 45 [both are -20 ]
4^2 - 2*4*9/2 + (9/2)^2 = 5^2 - 2*5*9/2 + (9/2)^2
[adding (9/2)^2 on both sides, rewriting 36 = 2*4*9/2, etc ]

(4 - 9/2)^2 = (5 - 9/2)^2
4 - 9/2 = 5 - 9/2 [ take square root of both sides ]
we get : 4 = 5 [adding 9/2 to each side]