To prove that an irrational number multiplied by an irrational number is rational, we can use the proof by contradiction method. This means assuming that the product of two irrational numbers is rational and then showing that this leads to a contradiction.
Yes, an example of two irrational numbers whose product is rational is √2 and √2. Both of these numbers are irrational, but when multiplied together, they equal 2, which is a rational number.
No, it is not possible for an irrational number multiplied by an irrational number to equal an irrational number. This is because multiplying any two irrational numbers will always result in a rational number.
This concept relates to the fundamental theorem of arithmetic because it shows that the product of two irrational numbers can always be expressed as a rational number, which follows the idea that all numbers can be broken down into prime factors.
Yes, this proof can be extended to other operations such as division or exponentiation. The same concept of using a proof by contradiction can be applied to show that the quotient or power of two irrational numbers will always result in a rational number.