Are irrational numbers closed under multiplication?

Understand the Problem

The question is asking whether the product of two irrational numbers is always an irrational number. To determine if irrational numbers are closed under multiplication, we need to investigate specific examples and mathematical properties related to irrational numbers and multiplication.

More Information

A set is said to be closed under an operation if performing that operation on members of the set always produces a member of the same set. For instance, the product of two irrational numbers might be a rational number (e.g., √2 * √2 = 2), showing that the set of irrational numbers is not closed under multiplication.

Tips

Common mistakes include assuming all operations will remain within the set if they do for some specific cases, and not testing sufficient examples.

Sources

• Irrational numbers - math.net
• The set of irrational number not closed under ordinary multiplication - math.stackexchange.com
• The set of irrational numbers is not closed under multiplication - byjus.com