## Infinity and zero

• 3.1k
"/" means division and "x" means multiplication

a) 0 / 4 = 0 and 4 x 0 = 0 + 0 + 0 + 0 = 0.....multiplication and division are inverse operations

b) 0 / n = 0 and n x 0 = 0 + 0 + 0 +...n times = 0

c) 0 / infinity = 0 and infinity x 0 = 0 + 0 + 0 +...infinity times = 0

but 0 = 0 / m as a rational number

so c) above becomes

d) 0 / infinity = 0 / m

e) if r / s = t / u then st = ur (cross multiplication) = s / r = u / t (dividing both sides by tr)

f) so r / s = t / u implies that s / r = u / t

g) using f) we see that d) becomes infinity / 0 = m / 0

h) we know from c) above that there are an infinite number of 0's in 0 so, infinity / 0 = infinity

i) So infinity / 0 = infinity = m / 0

k) therefore, any number m divided by 0 is infinity

Where is my error? Thanks for the help.

[EDIT] It seems that there are infinite 0's in 0 and not infinity. Sorry. My argument is wrong. Still I'd like you to find other errors in the argument. Thanks
• 2k
You can't divide by zero. Dividing by zero is just shorthand for the limit of a/b where b tends to 0. Whether a/b tends to infinity when b tends to zero depends entirely on a and b. EG, if a = x^2 and b=x then a/b tends to x^2 / x = x tends to 0. Or a=2x^2, b=x^2 similarly produces 2.
• 3.1k
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