Why 0/0 is undefined? As 1*0 = 1*0 then 1/1 = 0/0 which results in 0/0 = 1.

[Agent Smith]

Symbol for indeterminate is reverse question mark but my keyboard doesn't have that symbol.

Well, let's start from 0.

[imath]\frac{n}{0} = x, n \ne 0[/imath]
That means [imath]x(0) = n[/imath] has to be true, but [imath]\forall x(x(0) = 0)[/imath]. Ergo, [imath], n \ne 0, \frac{n}{0}[/imath] is undefined/meaningless.

[imath]\frac{n}{0} = y, n = 0[/imath]
[imath]\frac{0}{0} = y[/imath]. That means [imath]y(0) = 0[/imath]. But [imath]y[/imath] could be any number. We've dealt with nonequalities/inequalities. For example, [imath]y > 2x + 8[/imath]. Apply the same principle. [imath]\frac{0}{0} = y, y \in R[/imath] (ALL NUMBERS are SOLUTIONS to [imath]\frac{0}{0})[/imath]

You could say [imath]\frac{0}{0}[/imath] is (just) another way to express the set of real numbers, perhaps even maps to the complex plane, etc.

 

[khansaheb]

Symbol for indeterminate is reverse question mark but my keyboard doesn't have that symbol.

Are you talking about ∞ symbol?

That symbol can be created if have an extended MS keyboard (with NUM) pad.

Press down on ALT key then type 236 using the NUM keys.

 

[Agent Smith]

There is some confusion about the definitions:

Undefined: This usually means that there is no possible value that can be given, or that there is more than one value that it can logically be given.

Indeterminate: This usually means that there is not enough information to determine what the value should be.

If you are just talking about 0/0, then it is undefined. We can say that 0/0 is any number we like.

What you (apparently) found was an article about limits: when we see that
[imath]\displaystyle \lim_{x \to a} \dfrac{f(x)}{g(x)} \to \dfrac{0}{0}[/imath]

we say this is an "indeterminate form" because the limit does have some value (which may be infinite), we just have to do some work to figure out what it is for this particular limit.

For example:
[imath]\displaystyle \lim_{x \to 0} \dfrac{x^2 + 3x}{2x^2-x} = -3[/imath]

and
[imath]\displaystyle \lim_{x \to 0} \dfrac{sin(x)}{x} = 1[/imath]

despite both of them having the same indeterminate form, 0/0.

-Dan

@onkonovice