Question 155908
No irrational numbers are not closed under subtraction. Take this example {{{sqrt(2)-sqrt(2)=0}}}. Since {{{0=0/1}}}, this shows that 0 is rational (not irrational). So this means that subtraction is not closed over the irrationals (since we found a counter-example)





The same answer applies to division. Notice how {{{sqrt(2)/sqrt(2)=1}}} and {{{1=1/1}}}. So this shows us that division of irrational numbers can result in a rational number. So this also means that division is not a closed over the irrational numbers




Note: remember, the term "closed" just means that if you apply an operation to two numbers in a given set, then the result will be a number in that same set. For example, addition over the real numbers is closed since adding <b>any</b> two real numbers results in a real number.