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Is (1 / 2) / (2 / 0) defined?





Peterssidan
As we all know division by zero is not defined.

This expression contains a division by zero. Does that mean that the whole expression i undefined?

(1 / 2) / (2 / 0)

If we try to calculate the numerator (1 / 2) and denumerator (2 / 0) we end up with 0,5 divided by something that is undefined. There is obviously no way forward to get a sensible value out of this.

If we instead use the rule that says dividing by a fraction is the same as multiplying by the inverse we get the result zero.

(1 / 2) (0 / 2) = (1 0) / (2 2) = 0 / 4 = 0

So is this a valid/well defined expression or not?
badai
look at the original expression.

only, and only in linguistic can you re arrange you sentence.
kelseymh
Peterssidan wrote:
As we all know division by zero is not defined.


That depends on what you mean by "not defined." If that zero is literal, you are correct. If that zero is a limit, then the result may be well defined, but divergent, depending on the particular limit involved. Specifically:

lim (x -> 0+) 2/x = +infinity
lim (x -> 0-) 2/x = -infinity

The first expression has the limit approaching zero from the positive side, and the expression diverges to positive infinity. The second has the limit approaching zero from the negative side, and diverges to negative infinity.

You may see this graphically by plotting the function 1/x over the reals. There is one hyperbola in the first quadrant (the first expression) and one in the third quadrant (the second expression).

Quote:
This expression contains a division by zero. Does that mean that the whole expression i undefined?

(1 / 2) / (2 / 0)

If we try to calculate the numerator (1 / 2) and denumerator (2 / 0) we end up with 0,5 divided by something that is undefined. There is obviously no way forward to get a sensible value out of this.

If we instead use the rule that says dividing by a fraction is the same as multiplying by the inverse we get the result zero.

(1 / 2) (0 / 2) = (1 0) / (2 2) = 0 / 4 = 0

So is this a valid/well defined expression or not?


The second result is correct, provided you handle your limits correctly:

lim (x->0) (1/2)/(2/x) = lim(x->0) x/4 = 0

And if you have a directional limit, then you may write the final result as either +0 or -0 as needed.

Conveniently, in modern floating point arithmetic on your computer (IEEE-754), these sorts of limit results _are_ handled, with +/-0 and +/-infinity both having defined representations in floating point format, and with arithmetic computations which know how to propagate them through.
Peterssidan
Ah, yes limits (long time since I used them) but then I have to write that out lim (like you did) so the way I wrote it must be incorrect I guess. I can't say if it makes sense to use limits here because this is just an expression that I came up with.
loveandormoney
The answer iof the question si yes.
Best way to make an analysis of the lines is to use a matrix.
kelseymh
Peterssidan wrote:
Ah, yes limits (long time since I used them) but then I have to write that out lim (like you did) so the way I wrote it must be incorrect I guess. I can't say if it makes sense to use limits here because this is just an expression that I came up with.


Not incorrect, per se, just hard to pin down Smile You need to know how the zero was obtained (i.e., the full limit expression), because in more complex cases it can make a difference. For example, 0/0 is functionally undefined (not zero, not infinity, not one, not anything). But suppose the original expression was, let's say lim (x->0) x/x^2. Then we can simplify inside the limit to get lim (x->0) 1/x, and now we know this is +/-infinity.

Alternatively, supposed the original expression was lim (x->0) sin(x)/x. You can look up the details of doing this formally, but we can approximate -- for small values of x, sin(x) ~ x, so you end up with lim (x->0) x/x = 1.

The bottom line is that arithmetic involving zeroes or infinities is tricky Smile
Da Rossa
Yes, (1 / 2) / (2 / 0) is undefined. Think of it as a fraction consisting of a/b. a = 1/2 and b = 2/0. So b is not a Real number, therefore you can't have a/b.
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