Division by Zero: Is it Defined?

[dyn]

Hi.
x² = x³ / x but x² is defined for all x and equals zero at x=0 but what happens for x³ / x at x=o ? Is it defined at x=o ? Does it equal zero ? If not what is causing this anomaly ?
Thanks

 

[phinds]

N/0 is undefined regardless of N. If you assume otherwise, you can prove n=m where n and m are any arbitrary and different numbers.

 

[fresh_42]

Hi.
x² = x³ / x but x² is defined for all x and equals zero at x=0 but what happens for x³ / x at x=o ? Is it defined at x=o ? Does it equal zero ? If not what is causing this anomaly ?
Thanks

$x \longmapsto x$ is everywhere continuous, $x\longmapsto \dfrac{x^3}{x^2}$ is not; at $x=0\,.$

Although this is a removable singularity, it still is one, a gap. Algebraically division by zero isn't defined, simply because zero isn't part of any multiplicative group. The question never arises. It's like discussing the height of an apple tree on the moon.

 

[FactChecker]

When people write $x = \frac {x^3}{x}$ with no restriction that $x \ne 0$, they are being (understandably) careless. The proper way is to keep track of all those divisions by zero and make sure that the results are still legitimate when the simplified equations are used. Otherwise, rule those points out. In physical applications, the continuity and nice behavior of the reduced formula, $x$, at 0 makes it likely to also be valid at that point ($x = 0$).

 

[dyn]

So x = x³ / x² is not a correct statement on its own ? It needs the addition of the statement " x not equal to 0 " ?
I have seen the following statement in textbooks " xⁿ / x^m = x^n-m " with no mention of " x not equal to 0 ". Are they just being lazy and missing out the " x not equal to 0 " statement ?

 

[fresh_42]

So x = x³ / x² is not a correct statement on its own ? It needs the addition of the statement " x not equal to 0 " ?

Strictly, yes. But as it isn't defined for $x=0$ it is implicitly clear. As long as you don't want to write unnecessary additional lines, just leave it. Who writes $x\geq 0$ if he uses $\sqrt{x}\,?$ This is simply self-evident, resp. clear by context. But logically, the domain of $x$ needs to be mentioned in general, such that we know what the function really is. But in your post it was pretty clear what you meant even without it.

I have seen the following statement in textbooks " xⁿ / x^m = x^n-m " with no mention of " x not equal to 0 ". Are they just being lazy and missing out the " x not equal to 0 " statement ?

See above. It is simply not necessary as long as you don't write a book on logic. It's like mocking about a Pizza guy not telling you it's hot. However, if you talk about specific functions, you better say where and how they are defined. E.g. you could define
$$f(x) = \begin{cases}\dfrac{x^3}{x^2} \,&,\, x\neq 0 \\ 0\,&\,,x=0\end{cases}$$
or simply
$$
f(x)= \dfrac{x^3}{x^2}\; , \;x\neq 0
$$
which will be two different functions. So as always with written things, it depends on what you want to express. In post #1 and the example you gave it isn't necessary. Mocking about it is nit-picking.

 

[dyn]

Thank you for your replies.
I want to point out that I wasn't mocking about it ; I just wanted clarity. A lot of time some parts of maths seem like nit-picking to me but as plenty of people on here point out the finer points can be important.

 

[FactChecker]

Who writes $x\geq 0$ if he uses $\sqrt x$? This is simply self-evident, resp. clear by context.

I think that should be stated more carefully. Certainly, if $x = -y^2+y+5$ one would not use $\sqrt {x}$ without specifying that $-y^2+y+5 = x \ge 0$ and determining what the corresponding valid values of $y$ are.