Math Proofs

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3=0 (MAθ 2011, Open Proofs and Logic)

(1) x² + (x + 1) = 0	Since x = 0 is not a solution, divide (1) by x.
(2) x + 1 + ¹⁄_x = 0	Rearranging (2),
(3) x + 1 = ^-1⁄_x	Substituting (3) into (1),
(4) x² -¹⁄_x = 0	(4) can easily be solved (multiply both sides by x, noting x != 0).
(5) x = 1	Substituting (5) into original equation (1),
(6) 1² + 1 + 1 = 0
(7) 3 = 0.

1=0 (Wikipedia, Calculus)

(1) Let x = 1.
(2) ^d⁄_dx x = ^d⁄_dx 1
(3) 1 = 0.

1=3 (Wikipedia, Precalculus)

Noting e^πi = cos(π) + i sin(π) = -1 + 0i = -1,
and e^3πi = cos(3π) + i sin(3π) = -1 + 0i = -1,
(1) e^πi = e^3πi
(2) ln(e^πi) = ln(e^3πi)
(3) πi = 3πi
(4) 1 = 3.

2π=0 (Wikipedia, Algebra II)

Let x = 2π.
(1) sin(x) = 0	Taking arcsin of both sides,
(2) arcsin( sin(x) ) = arcsin(0)	Simplifying,
(3) x = 0	Substituting x=2π,
(4) 2π = 0.

2=1 (Wikipedia, Algebra I)

Let a and b be equal, nonzero quantities.
(1) a = b	Multiplying both sides by a,
(2) a² = ab	Subtracing b² from both sides,
(3) a² - b² = ab - b²	Factoring,
(4) (a + b)(a - b) = b(a - b)	Dividing both sides by (a - b),
(5) a + b = b	Substituting a = b,
(6) b + b = b
(7) 2b = b
(8) 2 = 1.