Question

Question #f08a8

Answer 1

Undefined, or Zero

Explanation:

By definition, anything being divided by zero (when it is in the denominator) is undefined (as it could be any value).

There is also the property that zero divided by anything is also zero.

This means that `0-:0` meets two of the properties, so the answer is both Undefined, and `0`

Answer 2

`0/0` is not defined. Any attempt to do so would require major changes in how we do arithmetic.

Explanation:

If we try to define `0/0 = c`, then we have

`2*0/0 = 2c` and also `2*0/0 = 2/1*0/0 = 0/0 = c`

We conclude that `2c = c` which implies that `c = 0`

If we try to define `0/0 = 0`, then we have:

`2/3 = 0 + 2/3 = 0/0+2/3`

`= (0 * 3)/(0 * 3) + (2 * 0)/(3 * 0)`

`= (0*3+2*0)/(3*0)`

`= (0 + 0)/0 = 0`

Therefore `2/3 = 0`

There is nothing special about `2/3` in this reasoning. We can also show that `5/7 = 0` and that `7 = 7/1 = 0` and that

for any number `x`, `x = x/1 = 0 + x/1 = * * * =0`

Our number system collapses to `0`.

This is not a useful result.

Answer 3

Another way of saying is, to take `alpha` being any negative or positive, real, imiginary or complex number.

We know that if `a*b=c`, then `a=c/b`, if we take `a=alpha`, and `b=0,`z then we get `0*alpha=0` or zero lots of anything is zero. By rearranging, we get `0/0=alpha`, where zero divided by zero, gives you every number possible; real, imaginary or complex.

Answer 4

Just to add some concrete examples of why `0//0` cannot just be assigned an arbitrary solution,

The following are all examples of the form `0//0`, taken directly from that WikiPedia Page on Indeterminate forms, so I take no credit for these examples:

Example 1

graph{x/x [-10, 10, -5, 5]}
`lim_(x rarr 0) x/x = 1`

Example 2

graph{x^2/x [-10, 10, -5, 5]}
`lim_(x rarr 0) x^2/x = 0`

Example 3

graph{sinx/x [-10, 10, -5, 5]}
`lim_(x rarr 0) sinx/x = 1`

Example 4

graph{(x-49)/(sqrt(x)-7) [-10, 50, -5, 20]}
`lim_(x rarr 49) (x-49)/(sqrt(x)-7) =14`

Example 5

graph{x/x^3 [-10, 10, -5, 30]}
`lim_(x rarr 49) x/x^3 rarr oo`

And a suitable example can easily be derived such that the for `0//0` takes any number we desire.