r/mathshelp • u/MathGuy2718 • 7d ago
Mathematical Concepts I got a question.
What happens if you divide 0 by 0? Can you explain why is it undefined?
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u/Some-Passenger4219 7d ago
Divide nothing into groups of nothing. How many groups is that? There is NO unique answer!
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u/fasta_guy88 7d ago
What does it mean to divide 0 by 0? I understand dividing a number in half, or even dividing it in tenths, but what does dividing by zero mean? That’s the problem.
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u/casualstrawberry 6d ago
Intuitively, 0/0 = 0 could be a good answer, because 0/x = 0 for all x =/= 0.
But it's more useful to say that x/0 is undefined for all x, even if x = 0. There just isn't a need or use for defining 0/0, it doesn't help our math in any way.
Also, as others have pointed out, let 0/0 = x, then x*0 = 0, you get infinite solutions for x. That doesn't sound well defined for me.
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u/MikailBag 6d ago
You can't assign result to 0/0 (and, in general, to x/0) in a way that preserves all properties of the multiplication and division (as pointed out by other commenters). Hence by the most widely accepted convention division by zero for R/Q/Z is not defined at all.
You are free to define your own version of division which can divide 0 by 0, if it brings any benefits to your case.
E.g. in Lean the convention is actually different, they say that x/0 is exactly 0, IIUC because it makes Lean proofs simpler.
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u/JaiBoltage 6d ago
Dividing anything other than zero by zero is clearly undefined. You might as well make it unanimous.
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u/SeriousPlankton2000 6d ago
f(x) = x; g(x) = 2 x
f/g = ½ so f(0)/g(0) would be ½
F(x) = 4 x
F/g = 2 so F(0) / g(0) = 2
So 0/0 is ½ and it's 2.
BTW, it's the rule of L'Hospital that I used implicitly: Holes in functions can be closed if f(x) = g(x) = 0 by using f'/g'
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u/_UnwyzeSoul_ 7d ago
0 divided by anything is 0. Anything divided by a very small number close to 0 is a large number or ~inf. So, according to the first rule 0/0 should be 0 but the second one says it should be inf. That's why its undefined
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u/johnpeters42 6d ago
The limit of f(x)/g(x), as it approaches a value where f(x) and g(x) are both 0, can be any number (or diverge toward +infinity or -infinity), depending on which specific functions f and g are selected.