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

  • Thread starter arbol
  • Start date
In summary: You can NOT put a decimal at a point. All you can do is to put a decimal in some place. That place is the "unit's" place. But since there is no point, there is no unit's place. So your "number" N has no meaning.In summary, the conversation discusses the concept of irrational numbers and how they are defined as numbers that cannot be expressed in the form of m/n, where m and n are integers and n is not equal to zero. The conversation also mentions the example of 1.234567891011... as an irrational number and how it can be represented as an infinite sequence without a fixed decimal point. However, the idea
  • #36
LukeD said:
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.
 
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  • #37
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.
 
  • #38
arbol said:
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?
 
  • #39
HallsofIvy said:
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.
 
  • #40
?? 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."
 
  • #41
HallsofIvy said:
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 [tex]X^X[/tex] 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!
 
Last edited:
<h2>What does it mean when someone says "There exists one number N"?</h2><p>When someone says "There exists one number N", they are stating that there is at least one specific number, denoted as N, that satisfies a given condition or set of conditions. This statement is often used in mathematical proofs to show that a solution or value exists for a particular problem.</p><h2>How do you determine the value of N in the statement "There exists one number N"?</h2><p>The value of N can be determined by analyzing the given conditions or constraints. In some cases, N may be explicitly stated or can be found by solving a mathematical equation or inequality. In other cases, N may be unknown and must be found through further investigation or experimentation.</p><h2>What is the significance of the statement "There exists one number N" in scientific research?</h2><p>The statement "There exists one number N" is often used in scientific research to show that a particular phenomenon or relationship exists. It can also be used to prove the existence of a solution or value in a mathematical or scientific problem. This statement is crucial in forming hypotheses and making conclusions based on data and evidence.</p><h2>Can there be more than one value of N that satisfies the statement "There exists one number N"?</h2><p>Yes, there can be more than one value of N that satisfies the statement "There exists one number N". This is because the statement only guarantees the existence of at least one value, but it does not limit the number of possible values that can satisfy the given conditions or constraints.</p><h2>What is the difference between "There exists one number N" and "For all numbers N"?</h2><p>The statement "There exists one number N" means that there is at least one specific number, denoted as N, that satisfies a given condition or set of conditions. On the other hand, the statement "For all numbers N" means that the given condition or set of conditions is true for every possible number, including N. In other words, "There exists one number N" is a more specific statement than "For all numbers N".</p>

What does it mean when someone says "There exists one number N"?

When someone says "There exists one number N", they are stating that there is at least one specific number, denoted as N, that satisfies a given condition or set of conditions. This statement is often used in mathematical proofs to show that a solution or value exists for a particular problem.

How do you determine the value of N in the statement "There exists one number N"?

The value of N can be determined by analyzing the given conditions or constraints. In some cases, N may be explicitly stated or can be found by solving a mathematical equation or inequality. In other cases, N may be unknown and must be found through further investigation or experimentation.

What is the significance of the statement "There exists one number N" in scientific research?

The statement "There exists one number N" is often used in scientific research to show that a particular phenomenon or relationship exists. It can also be used to prove the existence of a solution or value in a mathematical or scientific problem. This statement is crucial in forming hypotheses and making conclusions based on data and evidence.

Can there be more than one value of N that satisfies the statement "There exists one number N"?

Yes, there can be more than one value of N that satisfies the statement "There exists one number N". This is because the statement only guarantees the existence of at least one value, but it does not limit the number of possible values that can satisfy the given conditions or constraints.

What is the difference between "There exists one number N" and "For all numbers N"?

The statement "There exists one number N" means that there is at least one specific number, denoted as N, that satisfies a given condition or set of conditions. On the other hand, the statement "For all numbers N" means that the given condition or set of conditions is true for every possible number, including N. In other words, "There exists one number N" is a more specific statement than "For all numbers N".

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