Product Of Two Irrational Numbers
"The sum of 2 rational numbers is rational."
Past definition, a rational number can exist expressed equally a fraction with integer values in the numerator and denominator (denominator not zero). So, adding two rationals is the same every bit calculation two such fractions, which will result in another fraction of this aforementioned grade since integers are closed under improver and multiplication. Thus, adding two rational numbers produces some other rational number.
Proof:
"The product of 2 rational numbers is rational."
Again, by definition, a rational number tin can be expressed equally a fraction with integer values in the numerator and denominator (denominator not cipher). So, multiplying two rationals is the same every bit multiplying 2 such fractions, which will consequence in another fraction of this aforementioned form since integers are closed under multiplication. Thus, multiplying two rational numbers produces another rational number.
Proof:
Wait out! This next office gets tricky!!
"The sum of ii irrational numbers is SOMETIMES irrational."
The sum of two irrational numbers, in some cases, will exist irrational. Yet, if the irrational parts of the numbers have a nil sum (cancel each other out), the sum will be rational.
"The product of ii irrational numbers is SOMETIMES irrational."
The product of ii irrational numbers, in some cases, will be irrational. All the same, information technology is possible that some irrational numbers may multiply to course a rational product.
Product Of Two Irrational Numbers,
Source: https://mathbitsnotebook.com/Algebra1/RatIrratNumbers/RNRationalSumProduct.html
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