Is There a Number N That is Neither Rational nor Irrational?

[al-mahed]

In most cases, 0^0=1 is a sensible definition. A great many combinatorical identities rely on this.

Sorry, I didn't follow you...

I tought 0^0 didn't make any sense...

I know that if we take x^n such that x AND n --> 0, then x^n --> 1, and the two lateral limits converges to 1 in an ideal situation, but what of that since the "speed" of how x and n --> 0 is what matters?

 

[CRGreathouse]

I know that if we take x^n such that x AND n --> 0, then x^n --> 1, and the two lateral limits converges to 1 in an ideal situation, but what of that since the "speed" of how x and n --> 0 is what matters?

The rate of convergence has nothing to do with the final value.

 

[Hurkyl]

It is generally convenient to let the notation x^0 denote the constant polynomail 1. In fact, in terms of polynomial arithmetic, raising the polynomial x to the integer exponent 0 really does yield you the constant polynomial 1.

Since we are often interested in the functions associated to a polynomial or to a power series, the correct answer to "plug 0 into the polynomial x^0" is, in fact, 1. The order of operations is subtle here:

This is what happens:
Compute the polynomial exponent x^0 = 1
Evaluate the polynomial 1 at 0, getting 1.

This is not what happens:
Plug 0 in for x, getting the expression 0^0
Evaluate the exponentation 0^0, getting undefined.


In every case where I've seen it useful, that is what is meant by the convention "let 0^0 = 1".

 

[arbol]

It is generally convenient to let the notation x^0 denote the constant polynomail 1. In fact, in terms of polynomial arithmetic, raising the polynomial x to the integer exponent 0 really does yield you the constant polynomial 1.

Since we are often interested in the functions associated to a polynomial or to a power series, the correct answer to "plug 0 into the polynomial x^0" is, in fact, 1. The order of operations is subtle here:

This is what happens:
Compute the polynomial exponent x^0 = 1
Evaluate the polynomial 1 at 0, getting 1.

This is not what happens:
Plug 0 in for x, getting the expression 0^0
Evaluate the exponentation 0^0, getting undefined.


In every case where I've seen it useful, that is what is meant by the convention "let 0^0 = 1".

I think is best to leave 0^0 undefined since division by zero is undefined (unless you apply the limit concept).

 

[LukeD]

I think is best to leave 0^0 undefined since division by zero is undefined (unless you apply the limit concept).

No, it is best to leave it undefined in many cases precisely BECAUSE if you apply the limit concept you get that x^y cannot be extended continuously so that 0^0 has any value.

 

[arbol]

No, it is best to leave it undefined in many cases precisely BECAUSE if you apply the limit concept you get that x^y cannot be extended continuously so that 0^0 has any value.

Division by 0 is meaningless. Suppose that x is every element of the set R except 0, and y = 0.

If z = x/y, then x = y*z = 0, which is a contradiction. Therefore z is not properly defined when when say that

z = x/y.

For example,

it is not possible to write the following program:

1. x = 2
2. y = 0
3. z = x/y
4. print z

Here the output of the computer will normally be an error message at line 3.

Suppose that x = y = 0.

If z = x/y, then x = y*z = 0, where z is every element of the set R. Therefore z is not uniquely defined when we say that

z = x/y.

For example,

it is also not possible to write the following program:

1. x = 0
2. y = 0
3. z = x/y
4. print z

Again here the output of the computer will normally be an error message at line 3.

Suppose that x = y = 0**n = 0, where n is every element of the set R except 0.

If z = x/y, then x = y*z = 0, where z is every element of the set R, and z again is not uniquely defined when we say that

z = x/y.

For example,

it is also not possible to write the following program:

1. n = 2
2. x = y = 0**n
3. z = x/y
4. print z

Once again here the output of the computer will normally be an error message at line 3.

But suppose again that x = y = 0**n = 0, where n is every element of the set R except 0.

If x = y*z = 0, then we can define z = 1 and implicitly say (and it is understood) that z = x/y = 0**0 = 1. (This is a special case, where we have defined z = 1 and can implicitly say that z = x/y = 0**0 = 1.)

For example,

it is possible to write the following program:

1. n = 2
2. x = y = 0**n
3. z = 1
3. x = y*z
4. print x

Here the output of the computer is 0. Thus we can implicitly say, from x = y*z, that

z = x/y = 0**0 =1.

Consequently, suppose that x = y = 0**n = 1, where n = 0.

If z = x/y, then x = y*z = 1.

For example,

it is possible to write the following program:

1. x = 0**0
2. print x

Here the output of the computer is 1.

And if we write the following program:

1. x = y = 0**0
2. z = x/y
3. print z

the output of the computer will be 1.

 

[ramsey2879]

Arbol I just wanted to say that your last post is well thought out and very logical and correct. It is true though that some take the liberty to give their own definition of 0^0 if it suits their purposes.

 

[HallsofIvy]

It is necessary that N is not an interger, but it is one number.

You said, initially, that is was a natural number. All natural numbers are integers. So your first post was wrong?

 

[ramsey2879]

You said, initially, that is was a natural number. All natural numbers are integers. So your first post was wrong?

I agree that there is a contradiction there but I would rather that a mentor look to recognise that most everyone gains more insight as they interact with their surroundings and others and be a little more gentle in helping them to a greater understanding of things. You probably felt it necessary to point out the contradiction since Arbol has acted a little more knowing than is shown by his original posts and also was seemingly never willing to admit error. I don't mind as much and celebrate his better postings. If he has cause you to have bad feelings about him that is his loss and I would urge him to take note of your comment as a lessen not to be too conceited in the future and more willing to learn from someone who has more experience in the field.

 

[HallsofIvy]

?? I don't have any "bad feelings"- I simply pointed out that he was wrong. You seem to be under the impression that correcting an error is impolite. Certainly you wrote a very complementary response to his post that said, in essence, "computers say 0⁰= 1, therefore it is."

 

[ramsey2879]

You said, initially, that is was a natural number. All natural numbers are integers. So your first post was wrong?

On second thought, I would like Arbol to note that no number X can get close to infinity
since if X is a number then X^X is always less than infinity. Since others have shown that N as defined in his first post is infinity, I would like Arbol's response as N is neither a rational or irrational number!