SOLUTION: Which subsets of the real number system are closed under division by a nonzero number?

Question 1196050: Which subsets of the real number system are closed under division by a nonzero number?
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) (Show Source): You can put this solution on YOUR website!

Non-zero rational number sets is closed under division.
To find : Closed under division from the following sets.

(a) let’s check Natural numbers :
Natural numbers starts from {1, 2, 3, ......., n} numbers.
To find whether natural numbers is closed under division, consider an example :
Assume two values like and .
Divide and =>
Resultant is a natural number.
So, numbers are under division.

(b) let’s check Non - Zero integers :
Non-Zero integers which have positive and negative values.
To find whether non-zero integer is closed under division, consider an example :
Assume two non-zero values and.
Divide them, which gives
Resultant value is which is an .
That is, non-zero are under division

(c) let’s check Irrational numbers :
Irrational numbers, a number cannot be represented as fraction.
Eg : and
Divide the values, gives .
" ", which is an numbers.
So, numbers are under division.

(d) let’s check Non - Zero rational numbers :
Non zero rational numbers like , , , and so on.
To find whether non zero rational numbers is closed under division, consider an example :
Assume two non zero rational numbers and .
Dividing them we get, .
Resultant value is the numbers.
Hence, non zero closed}}} under division.

Therefore, non-zero are under .

Answer by ikleyn(52750) (Show Source): You can put this solution on YOUR website!
.
Which subsets of the real number system are closed under division by a nonzero number?
~~~~~~~~~~~~~~~~

The meaning of your request is really UNCLEAR.

It is written in so mathematically illiterate way, that I should spend half an hour (or more) to explain WHY it is so,
and explain HOW it SHOULD BE, which (time) I definitely do not have to spend it for nothing.

When a person comes with such or similar question, he/she should have some preliminary pre-requisites
and formulate his/her question correctly in the frame of these pre-requisites, in mathematically correct form.

Then he/she may expect to get an adequate answer.

///////////////

Actually, the field of real numbers has INFINITELY MANY sub-fields.

They are sub-fields - Q (rational numbers), - (an extension of Q with irrationality ), - (an extension of Q with irrationality ), - (an extension of Q with irrationality ), . . . and infinitely many other sub-fields of the form , where positive integer number n is not a perfect square, . . . - (an extension of Q with irrationalities and ), - (an extension of Q with irrationalities and ), - (an extension of Q with irrationalities and ), . . . and infinitely many other sub-fields of the form , where m and n are different non-square positive integer numbers, . . . - (an extension of Q with irrationality ), - (an extension of Q with irrationality ), - (an extension of Q with irrationality ), . . . and infinitely many other sub-fields of the form , where integer number n is not a perfect cube, . . . and so on and so on . . . - - - infinitely many others.

Each of these sub-fields is a sub-set of real numbers, closed relative addition, subtraction,
multiplication and division by a non-zero elements of these sub-fields.

By the way, when in the school, you learn about getting rid of irrationality in denominators,
you actually learn that the extension is a field over Q.

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