Mathematics

Algebra: One Equals Zero

A Simple Proof That 1= 0

a = b
a² = ab
a² - b² = ab - b²
(a + b)(a - b) = b(a - b)
a + b = b
b + b = b
2b = b
2 = 1
1 = 0

An Alternate Proof

-1/1 = 1/-1
sqrt(-1/1) = sqrt(1/-1)
sqrt(-1)/sqrt(1) = sqrt(1)/sqrt(-1)
i/1 = 1/i
i / 2 = 1 / (2i)
i/2 + 3/(2i) = 1/(2i) + 3/(2i)
i (i/2 + 3/(2i) ) = i ( 1/(2i) + 3/(2i) )
i^2/2 + 3i/2i = i/2i +3i/2i
(-1)/2 + 3/2 = 1/2 + 3/2
1 = 2
0 = 1

Another Proof

a = b
a² = ab
a² + a² = a² + ab
2a² = a² + ab
2a² - 2ab = a² + ab - 2ab
2a² - 2ab = a² - ab
2(a² - ab) = 1(a² - ab)
2 = 1
1 = 0

And Another

-2 = -2
4 - 6 = 1 - 3
4 - 6 + 9/4 = 1 - 3 + 9/4
(2 - 3/2)² = (1 - 3/2)²
2 - 3/2 = 1 - 3/2
2 = 1
1 = 0

For the Unconvinced

(n + 1)^2 = n^2 + 2n + 1
(n + 1)^2 - (2n + 1) = n^2
(n + 1)^2 - (n + 1)(2n + 1) + ¼(2n + 1)^2 = n^2 - n(2n + 1) + ¼(2n + 1)^2
[(n + 1) - ½(2n + 1)]^2 = [(n - ½(2n + 1)]^2
(n + 1) - ½(2n + 1) = n - ½(2n + 1)
n + 1 = n
1 = 0