Why does 0/0 or 1/0 or 0^0= undefined and why is 0/1=0?

Question

Why is 0/0 or 1/0 or 0^0= undefined and why is 0/1=0

Answer 1

For 0/0, let us 'solve' 0*x = 0. Note that this is true for all x. 'By dividing both sides by 0,' we arrive that 0/0 = x. So 0/0 is any number we want. Hence it cannot be defined.

For 0/1, (by definition of division) recall that 0/1 means 0*(1/1) which is 0*1, and 0 times any finite number is 0. So 0/1 = 0.

For 1/0, let us 'solve' 0*x=1. Note that this is false for all x. 'By dividing both sides by 0', we arrive that 1/0 = x. But since there is no possibility for x, there is no possibility for 1/0. Hence we cannot define 1/0 as any number.

For 0^0, we require a bit of cleverness. First, for any y, note that if y^0 is definied, it must equal 1 (i.e., anything raised to the 0th power is 1). So if 0^0 is defined, it must be 1. Now, suppose 0^0 is defined, then 0^0=1, then by 'rasing both sides to the 1/0 power' 0 = 1^(1/0). However, 1 raised to any power is 1. So 0 = 1. Therefore the assumed hypothesis that 0^0 is defined must be false. So 0^0 is undefined.

Except for 0/1, these techniques are technically flawed. However, the general idea is correct: you can make 0/0 anything you want, 0^0 any nonnegative number, and there is no number that can be 1/0. A rigorous and precise explaination why 0/0, 1/0 and 0^0 are all undefined requires very sophisticated mathematics (specifically Advanced Calculus or Real Analysis). The explainations provided here should suffice for people that are familar with beginning algebra (except for 0^0 which requires more intermediate algebra.)

Answer 2

Any number divided by 0 is undefined, because, quite simply, it's the way division was defined to be. Allowing division by zero creates a multitude of problems.

0^0 is undefined because 0 to the power of anything is 0, yet anything to the power of 0 is equal to 1. Which one wins? I'm not entirely sure why 0^0 is undefined, but it is definitely on the list of indeterminate forms of solving limits in Calculus.

Answer 3

You can not divide anything by 0 because there is nothing to divide by therefore you get undefined.

0/1 = 0 because 0 has no factors. Like my elementary teacher use to say give me 1 zero now divide that zero by 1 you still end up with 0.

Answer 4

0/0 depends on which zero is bigger, traditionally it is infinity take the derrivative of the top and bottom and solve it. for the first one ... you can do it two ways, replace the zero with a variable, then distribute the exponent to the top and bottom, which makes 1/1 which, yes is 1. but if you dont do the variable thing, then 2/0 is undefined, and you can't put an exponent on something that is strictly undefined. Basically, from a standard algebra standpoint, these are impossible equations. (My algebra teacher said, that when you divide by zero, it's like setting off an A-bomb all over your math homework!) However, with higher mathematics, like abstract algebra, or linear equations, maybe even with only a bit of calculus, you could argue one way or another. ... ::Shrug::

Answer 5

0/0 is one because you can cancel out the 0's in the same way you do in 0^0, so say you have 0^0. That means you have (0 X 0)/(0 X 0)
Cancel out each of the 0's & you get (1X1)/(1X1)=1. However 1/0 is undefined because if you have 1 object distributed among 0 people, no people have any objects, but it is still there (so it cant be 0) and so its just sitting there, causing it to be undefined. but you can have a multiplicative inverse of 1/0(0/1) and end up as 1.

Answer 6

0/0 is undefined, as is 0^0. However, you can usefully define 1/0 as "infinity", rather than leave it undefined.

Answer 7

whenever the denomentator is zero then the answer is undefined.

lets think of x/y as:
i have x out of y apples.

if you have 0/1 then you have zero apples (out of 1 apple in the basket). so you have zero apples.

BUT, if you had 1/0 then you have 1 apple (out of zero apples in the basket) -- this makes no sense so there is no answer! it is undefined.
if it is 0/0 then you have zero apples... but it was out of an origional zero apples from the basket! so it is also undefined.

general rule tho -- whenever there is a zero in the denomenator (the bottom of the fraction) then the answer is undefined, and whenever the zero is in the numerator (the top of the fraction) then the answer is going to be 1.


done.

Answer 8

it has to do with the concept of zero
if you divide at the rate of zero any thing you can divide it among1,10987,374896978379438,and still you will have your zero in tact
that is why anything divided by 0 is indeterminate
0^0 doesn^t make sense at all by the concept of both zero and power
if you divide nothing among n people each will get nothing and so 0/n=0