To address the question "The product of two different irrational numbers is always," let's examine the nature of irrational numbers and their products.
1. Definition of Irrational Numbers: An irrational number is a number that cannot be expressed as a fraction [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. Irrational numbers have non-repeating, non-terminating decimal expansions. Examples of irrational numbers include [tex]\( \sqrt{2} \)[/tex], [tex]\( \sqrt{3} \)[/tex], and [tex]\( \pi \)[/tex].
2. Exploring Products of Irrational Numbers:
- Case 1: Producing a Rational Number
- Consider [tex]\( \sqrt{2} \)[/tex] and [tex]\( \sqrt{2} \)[/tex].
- Their product is [tex]\( \sqrt{2} \times \sqrt{2} = 2 \)[/tex], which is a rational number.
- Case 2: Producing an Irrational Number
- Consider [tex]\( \sqrt{2} \)[/tex] and [tex]\( \sqrt{3} \)[/tex].
- Their product is [tex]\( \sqrt{2} \times \sqrt{3} = \sqrt{6} \)[/tex], which is an irrational number.
3. Conclusion: Since the product of two different irrational numbers can result in either a rational or an irrational number, it is not consistent. Thus, the product of two different irrational numbers is not definitively always one type over the other.
Therefore, the product of two different irrational numbers is not always irrational nor rational. There is no single consistent answer.
