Option C: [tex]\sqrt{10} +(-\sqrt{10})=0[/tex] is an example of why irrational numbers are not closed under addition.
Explanation:
For a irrational number to be closed under addition, the sum of two numbers of an irrational number must also be an irrational.
Option A : [tex]\sqrt{4} +\sqrt{4} =2+2=4[/tex] and 4 is not irrational.
From the expression, we can see that [tex]\sqrt{4}[/tex] is a rational number because it results in a rational number. That is, [tex]$\sqrt{4}=2$[/tex]
Thus, Option A is not the correct answer.
Option B : [tex]\frac{1}{2} +\frac{1}{2} =1[/tex] and 1 is not irrational.
From the expression, we can see that [tex]$\frac{1}{2}$[/tex] is a rational number.
Hence, the addition of two rational numbers results in a rational number.
Thus, Option B is not the correct answer.
Option C : [tex]\sqrt{10} +(-\sqrt{10})=0[/tex] and 0 is not irrational.
From the expression, we can see that [tex]$\sqrt{10}$[/tex] is an irrational number because it is a non - terminating decimal number.
Hence, the addition of two irrational number is a rational number.
Therefore, the irrational numbers are not closed under addition because the addition of irrational numbers does not result in a irrational number.
Thus, Option C is the correct answer.
Option D : [tex]-3+3=0[/tex] and 0 is not irrational.
From the expression, we can see that 3 is a rational number.
Hence, the addition of two rational numbers results in a rational number.
Thus, Option D is not the correct answer.
Answer C:
Explanation: For a irrational number to be closed under addition, the sum of two numbers of an irrational number must also be an irrational.
