Can $\frac{a}{0}$ be defined or solved?
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2Can you find a $b$ such that $b \cdot 0 = a,$? – dxiv Jan 17 '17 at 04:59
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No. No, it can not. – fleablood Jan 17 '17 at 05:45
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3I'm not sure what it would even mean for $\frac{1}{2}$ to be solved. – Jan 17 '17 at 06:10
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2You do not solve numbers. You solve problems, questions, or difficulties. – DanielWainfleet Jan 17 '17 at 06:41
3 Answers
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Let one equation -
$\frac40 = x$
Then $4 = 0 \times x$
But no value would work for x because 0 times any number is 0. So division by zero doesn't work.
Kanwaljit Singh
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Algebraically, the projective numbers are convenient, which adds a single point at $\infty$ (in particular, $\infty = -\infty$ in this number system), and defines $a/0 = \infty$ whenever $a \neq 0$.
You have to keep in mind, however, that $0/0$ is still undefined, and so are $\infty/\infty$, $\infty + \infty$, and $\infty - \infty$.
Projective coordinates may be enlightening; let $(a:b)$ (for real $a,b$) denote the projective number $a/b$. Alternatively, we can define coordinates without reference to $\infty$, by saying that $(a:b) = (c:d)$ if and only if $ad=bc$, and then define $\infty = (1:0)$.
Then, the elementary arithmetic operations are
$$ (a:b) + (c:d) = (ad+bc : bd)$$ $$ (a:b) - (c:d) = (ad-bc : bd)$$ $$ (a:b) \cdot (c:d) = (ac:bd) $$ $$ (a:b) / (c:d) = (ad:bc) $$ $$ (a:b)^{-1} = (b:a) $$
In projective coordinates, $(0:0)$ is not allowed, so any arithmetic operation that would give that result is instead left undefined.
There is an even more general theory called "wheel theory" that would let us further extend to allow $0/0$ to be a number as well. They're kinda neat, but I am not aware of wheels being used beyond the work of their inventor.
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This can be better understood using Limits. Consider a function (n/x), where n is a positive whole number constant, and x is a variable. When you decrease x to a number so small (say E-20), the function obtains a very large value. As x -> 0, the limit of the function is a very very large number. However, this number is undefined, because there is no clear way to express it (people use infinity to define it. But note that infinity is a concept, not a number).
There are some cases where a / 0 is defined, such as a Riemann Sphere, but the structures in these cases do not follow every rule of arithmetic.
