Is division closed under rational numbers and if yes, what if I divide 0.5 by 0, I will get an undefinined answer. So is division closed under rational numbers .

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Solution

No, rational numbers are not closed under division. For example, let us take two rational numbers 2 and 0. Now, 2 / 0 is not defined. As division of two rational numbers 2 and 0 is not a rational number, so division of two rational number is not closed.
The rational numbers are not closed with respect to division because 0 is a rational number but dividing any other rational number by 0 does NOT produce a rational number.

Proof: if it did, then multiplying the result by 0 would produce the original rational number, but any number multiplied by 0 is 0 so the original rational number could only have been 00 and not any other rational number.

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Similar questions

Q. (i) Are rational numbers always closed under division?
(ii) Are rational numbers always commutative under division?
(iii) Are rational numbers always associative under division?
(iv) Can we divide 1 by 0?

Q. Manu while having conversation with Tanu.

Manu: Rational numbers are closed under division because we get a rational number as the resultant:

\frac{3}{5} \div \frac{7}{5} = \frac{3}{7}

Tanu: Rational numbers are NOT closed under division because dividing any number by zero is undefined.

\frac{3}{5} \div 0

is undefined

Who is correct?

Which of the following is not true?
(i) Whole numbers are closed under subtraction.
(ii) Integers are not closed under division.
(iii) Rational numbers are closed under division.

Question 4
Which of the following is not true?
(a) rational numbers are closed under addition.
(b) rational numbers are closed under subtraction.
(c) rational numbers are closed under multiplication.
(d) rational numbers are closed under division.

Q. Write 'T' for true and 'F' for false for each of the following:
(i) Rational numbers are always closed under subtraction.
(ii) Rational numbers are aways closed under division.
(iii) 1 ÷ 0 = 0.
(iv) Subtraction is commutative on rational numbers.
(v)

- (\frac{- 7}{8}) = \frac{7}{8} .

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