Question
Give one example each to show that the rational numbers are closed under addition, subtraction and multiplication. Are rational numbers closed under division? Give two examples in support of your answer.
✓
Answer
We know that rational numbers are closed under addition, subtraction and multiplication. We can understand this from the following examples:
Rational numbers are closed under
Addition
e.g. $\frac{4}{7}+\frac{1}{2}=\frac{8+7}{14}=\frac{15}{14}$ which is a rational number.
Subtraction
e.g. $\frac{4}{7}-\frac{1}{2}=\frac{8-7}{14}=\frac{1}{14}$ which is a rational number.
Multiplication
e.g. $\frac{4}{7} \times \frac{1}{2}=\frac{4}{14}=\frac{2}{7}$ which is a rational number.
But rational numbers are not closed under division. If zero is excluded from the collection of rational numbers, then we can say that rational numbers are closed under division.
Now, we see the examples given below
$\frac{4}{7}+\frac{1}{2}=\frac{4}{7} \times 2=\frac{8}{7}$ which is a rational number.
But $\frac{4}{7}\div0=\frac{4}{7} \times \frac{1}{0}$
which is not defined and so, it is not a rational number.
Also, $\frac{1}{2}\div0=\frac{1}{2} \times \frac{1}{0}$,
which is not defined and also not a rational number.
Rational numbers are closed under
Addition
e.g. $\frac{4}{7}+\frac{1}{2}=\frac{8+7}{14}=\frac{15}{14}$ which is a rational number.
Subtraction
e.g. $\frac{4}{7}-\frac{1}{2}=\frac{8-7}{14}=\frac{1}{14}$ which is a rational number.
Multiplication
e.g. $\frac{4}{7} \times \frac{1}{2}=\frac{4}{14}=\frac{2}{7}$ which is a rational number.
But rational numbers are not closed under division. If zero is excluded from the collection of rational numbers, then we can say that rational numbers are closed under division.
Now, we see the examples given below
$\frac{4}{7}+\frac{1}{2}=\frac{4}{7} \times 2=\frac{8}{7}$ which is a rational number.
But $\frac{4}{7}\div0=\frac{4}{7} \times \frac{1}{0}$
which is not defined and so, it is not a rational number.
Also, $\frac{1}{2}\div0=\frac{1}{2} \times \frac{1}{0}$,
which is not defined and also not a rational number.
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