Exclude natural numbers in the set of rational numbers (using math symbols)

[Nemanjavuk69]

Hello

I am curious how would one write, with mathematics notaions, that $$\frac{k}{2}$$ HAS to be a decimal number.
I know rational numbers is all the natural numbers (and negative) plus the... well, rational numbers, however, I want to exclude the natural numbers.
$$\frac{k}{2}$$ can therefore not be 1, 2, 3... etc.
however, $$\frac{k}{2}$$ can be 1.5, 2.6, 3.33... etc.
With mathematics notation, how would one write that?

My guess would be something like this, but I am not sure.

$$\frac{k}{2} \exists \in Q \land (\notin \N)$$

 

[pka]

Exclude natural numbers in the set of rational numbers (using math symbols)​

Taking the title of the heard literally the answer is: $\mathbb{Q}\setminus\mathbb{N}$


$$

 

[Nemanjavuk69]

Exclude natural numbers in the set of rational numbers (using math symbols)​

Taking the title of the heard literally the answer is: $\mathbb{Q}\setminus\mathbb{N}$


$$

Hello, thank you very much for your answer. Just to be sure, would $\frac{k}{2} \exist \in \frac{Q}{\N}$ literally mean, that $\frac{k}{2}$ has to be a decimal point number?

 

[pka]

Hello, thank you very much for your answer. Just to be sure, would $\frac{k}{2} \exist \in \frac{Q}{\N}$ literally mean, that $\frac{k}{2}$ has to be a decimal point number?

Sorry, but your notation is completely & utterly nonsense!

 

[Nemanjavuk69]

Sorry, but your notation is completely & utterly nonsense!

I am sorry, but that is exactly what I am asking help with. I don't know how much more precise I can be in my title or in my description. My question is not whether or not I am right, but HOW to ACTUALLY write it using mathematics notations.

 

[pka]

I am sorry, but that is exactly what I am asking help with. I don't know how much more precise I can be in my title or in my description. My question is not whether or not I am right, but HOW to ACTUALLY write it using mathematics notations.

Do you not understand that I have given you the answer.$\mathbb{Q}\setminus\mathbb{N}$
To write $t\in\mathbb{Q}\setminus\mathbb{N}$ means that $t$ is a rational number that is not a natural number.
It is standard notation $A\setminus B$ is the set of all elements of $A$ that are not in set $B$

$$

 

[topsquark]

Hello

I am curious how would one write, with mathematics notaions, that $$\frac{k}{2}$$ HAS to be a decimal number.
I know rational numbers is all the natural numbers (and negative) plus the... well, rational numbers, however, I want to exclude the natural numbers.
$$\frac{k}{2}$$ can therefore not be 1, 2, 3... etc.
however, $$\frac{k}{2}$$ can be 1.5, 2.6, 3.33... etc.
With mathematics notation, how would one write that?

My guess would be something like this, but I am not sure.

$$\frac{k}{2} \exists \in Q \land (\notin \N)$$

I have to ask...

What you are literally asking for is the answer that pka gave you. But your example makes me wonder. You do understand that 1.5 is a rational number, right? Are you asking for an irrational value for k/2 or are you just looking for non-integer values of k/2?

-Dan

 

[Deleted member 4993]

I have to ask...

What you are literally asking for is the answer that pka gave you. But your example makes me wonder. You do understand that 1.5 is a rational number, right? Are you asking for an irrational value for k/2 or are you just looking for non-integer values of k/2?

-Dan

I think op wants exclude "integer" answers.

 

[Deleted member 4993]

Hello

I am curious how would one write, with mathematics notaions, that $$\frac{k}{2}$$ HAS to be a decimal number.
I know rational numbers is all the natural numbers (and negative) plus the... well, rational numbers, however, I want to exclude the natural numbers.
$$\frac{k}{2}$$ can therefore not be 1, 2, 3... etc.
however, $$\frac{k}{2}$$ can be 1.5, 2.6, 3.33... etc.
With mathematics notation, how would one write that?

My guess would be something like this, but I am not sure.

$$\frac{k}{2} \exists \in Q \land (\notin \N)$$

Do you know the difference between "integers" and "rational numbers" and "natural numbers"? Please define those three terms.

 

[Dr.Peterson]

My guess would be something like this, but I am not sure.

$$\frac{k}{2} \exists \in Q \land (\notin \N)$$

Although you say you are asking for the correct way to say it, it is also important to know the correct way to use the symbols you used, and part of that is to know why what you wrote is, in fact, nonsense.

A literal translation of what you wrote would be

k/2 there exists is an element of the set of rational numbers and is not an element of the set of natural numbers.​

This is bad grammar in English; but it is even worse in symbols, as we aren't allowed to use implied subjects, as in "and is not an element of".

The existential quantifier is not needed here, as you are not claiming that something exists, but merely describing something that presumably has been previously defined (k). Here is a corrected version of what you wrote:

$$\frac{k}{2} \in Q \land \frac{k}{2}\notin \N$$
But as you've been told, there is a better way that avoids the need to name the element twice:

$$\frac{k}{2} \in\mathbb{Q}\setminus\mathbb{N}$$
See here and here.

There is another issue that is unclear: In what sense do you want to say that k/2 "has to be a decimal number". Can you explain the context? This might affect how one would say it. (For example, is this a conclusion you are making, or a premise of an argument, or something else?) Of course, "decimal number" is rather unclear, and I suppose we are assuming you mean primarily "not an integer"; but is there anything requiring it to be a rational number, and not just any non-integer real number? Pi is just as much a "decimal number" as 3.333... is.

 

[Otis]

Pi is just as much a "decimal number" as 3.333... is.

The fact that we can't write it out notwithstanding, of course.

$\;$

 

[Nemanjavuk69]

Although you say you are asking for the correct way to say it, it is also important to know the correct way to use the symbols you used, and part of that is to know why what you wrote is, in fact, nonsense.

A literal translation of what you wrote would be

k/2 there exists is an element of the set of rational numbers and is not an element of the set of natural numbers.​

This is bad grammar in English; but it is even worse in symbols, as we aren't allowed to use implied subjects, as in "and is not an element of".

The existential quantifier is not needed here, as you are not claiming that something exists, but merely describing something that presumably has been previously defined (k). Here is a corrected version of what you wrote:

$$\frac{k}{2} \in Q \land \frac{k}{2}\notin \N$$
But as you've been told, there is a better way that avoids the need to name the element twice:

$$\frac{k}{2} \in\mathbb{Q}\setminus\mathbb{N}$$
See here and here.

There is another issue that is unclear: In what sense do you want to say that k/2 "has to be a decimal number". Can you explain the context? This might affect how one would say it. (For example, is this a conclusion you are making, or a premise of an argument, or something else?) Of course, "decimal number" is rather unclear, and I suppose we are assuming you mean primarily "not an integer"; but is there anything requiring it to be a rational number, and not just any non-integer real number? Pi is just as much a "decimal number" as 3.333... is.

Hello

Thanks for your reply and thank you for the ellaboration. As I stated earlier, my question was not if what I wrote was nonsense or not, in fact it is what I am needing help with writing. I didn't know it was nonsense, hence why it was my "guess", I was only looking for the complete way of typing it using math notations.

After seeing other people question my post, I can confirm that I am looking for non-integer values. With other words $\frac{k}{2}$ has to be ANY number where there is a "." even if that means Pi, 3.33, 1.5 or something else. Where I am from, we call this simply decimal numbers, since there is a decimal point ".". I think this is just a language barrier, and for that, I apologize I should had been more precise to save you time.

After @pka wrote $t \in \frac{\mathbb{Q}}{\N}$ as his second comment, I think I know what is meant now, but also have to be sure. Does $\frac{k}{2} \in \frac{\mathbb{Q}}{\N}$ mean that "$\frac{k}{2}$ exists in the set of rational numbers excluding the natural numbers"? Or put in a more simple term "$\frac{k}{2}$ has to be the value of a non-integer"?

 

[Dr.Peterson]

Thanks for your reply and thank you for the ellaboration. As I stated earlier, my question was not if what I wrote was nonsense or not, in fact it is what I am needing help with writing. I didn't know it was nonsense, hence why it was my "guess", I was only looking for the complete way of typing it using math notations.

If your goal includes learning to write mathematical notation, then you should want to understand its "grammar" in general, and therefore, just as in learning English, you would want to know more than just how to say one thing correctly. If I were learning a language, I would want to know if something I tried to say made no sense, in order to avoid it! But we can't force you to want more than you want.

After seeing other people question my post, I can confirm that I am looking for non-integer values. With other words $\frac{k}{2}$ has to be ANY number where there is a "." even if that means Pi, 3.33, 1.5 or something else. Where I am from, we call this simply decimal numbers, since there is a decimal point ".". I think this is just a language barrier, and for that, I apologize I should had been more precise to save you time.

It appears, then, that your real question does not involve rational numbers at all, since pi is not a rational number! This is why mathematics has precisely defined terms, and why you need to clearly state exactly what you mean in order to translate it into symbols.

It is possible that you are assuming that, since your expression is written as a fraction, it must be rational. But a rational number is any number that can be written as a fraction with integers in the numerator and denominator; numbers like $\frac{\pi}{2}$ are not rational. It is very common (among English speakers) to use the word "decimal" in ways that don't fit the proper mathematical definition; that is why we need to know your informal usage of the term, and why "decimal number" is not used formally in mathematics.

After @pka wrote $t \in \frac{\mathbb{Q}}{\N}$ as his second comment, I think I know what is meant now, but also have to be sure. Does $\frac{k}{2} \in \frac{\mathbb{Q}}{\N}$ mean that "$\frac{k}{2}$ exists in the set of rational numbers excluding the natural numbers"? Or put in a more simple term "$\frac{k}{2}$ has to be the value of a non-integer"?

The set difference notation is never written in vertical form as you are doing; that is why I provided two links to demonstrate proper usage to you. (In fact, the slash we use is not the same "forward slash" used for fractions at all!)

Since it appears you don't require your expression to be a rational number, the proper statement will be

$$\frac{k}{2} \in\mathbb{R}\setminus\mathbb{N}$$ or $$\frac{k}{2} \in\mathbb{R}-\mathbb{N}$$
where R means Real numbers. Or you could merely say, $$\frac{k}{2} \notin\mathbb{N}$$

 

[Nemanjavuk69]

If your goal includes learning to write mathematical notation, then you should want to understand its "grammar" in general, and therefore, just as in learning English, you would want to know more than just how to say one thing correctly. If I were learning a language, I would want to know if something I tried to say made no sense, in order to avoid it! But we can't force you to want more than you want.

It appears, then, that your real question does not involve rational numbers at all, since pi is not a rational number! This is why mathematics has precisely defined terms, and why you need to clearly state exactly what you mean in order to translate it into symbols.

It is possible that you are assuming that, since your expression is written as a fraction, it must be rational. But a rational number is any number that can be written as a fraction with integers in the numerator and denominator; numbers like $\frac{\pi}{2}$ are not rational. It is very common (among English speakers) to use the word "decimal" in ways that don't fit the proper mathematical definition; that is why we need to know your informal usage of the term, and why "decimal number" is not used formally in mathematics.

The set difference notation is never written in vertical form as you are doing; that is why I provided two links to demonstrate proper usage to you. (In fact, the slash we use is not the same "forward slash" used for fractions at all!)

Since it appears you don't require your expression to be a rational number, the proper statement will be

$$\frac{k}{2} \in\mathbb{R}\setminus\mathbb{N}$$ or $$\frac{k}{2} \in\mathbb{R}-\mathbb{N}$$
where R means Real numbers. Or you could merely say, $$\frac{k}{2} \notin\mathbb{N}$$

Hello again

Thank you so much for providing an elaborate answer. It makes a lot of sense. I am only looking for any answer where $\frac{k}{2}$ is a non-integer. I see now that we can write it as $\frac{k}{2} \in \mathbb{R} \setminus \N$ or as $\frac{k}{2} \in \mathbb{R} - \N$

What if, we define k to ONLY be a natural number would $\frac{k}{2} \in \mathbb{Q} \setminus \N$ or $\frac{k}{2} \in \mathbb{Q} - \N$

also be correct, since k can only be whole integers like 1, 2, 3... etc?

 

[Dr.Peterson]

What if, we define k to ONLY be a natural number would $\frac{k}{2} \in \mathbb{Q} \setminus \N$ or $\frac{k}{2} \in \mathbb{Q} - \N$

also be correct, since k can only be whole integers like 1, 2, 3... etc?

Yes, if k is a natural number, then k/2 will be rational.

But you've told us nothing about the context, so there may be other important adjustments to make, as I stated previously. If you told us more, we could probably say more. In particular, I am still unsure of what you mean by "HAS to be"; what we are saying means merely "is".

 

[Nemanjavuk69]

Yes, if k is a natural number, then k/2 will be rational.

But you've told us nothing about the context, so there may be other important adjustments to make, as I stated previously. If you told us more, we could probably say more. In particular, I am still unsure of what you mean by "HAS to be"; what we are saying means merely "is".

Don't worry, there are no more important adjustments. This was just purely curiosity that peaked in me.

Thanks once again and may you have a wonderful weekend!