Recall that a rational number is a number that can be written as , where and are integers.
We will explore some properties of rational numbers.
1) The sum of 2 rational numbers is always rational.
Proof: Let the 2 numbers be and , where are integers. Then, their sum is , and and are both integers. Hence, this number is rational.
2) The product of 2 irrational numbers does not need to be irrational.
Proof: In the previous post, we showed the is irrational. The product of and is 2, which is rational.
3) The sum of a rational number and an irrational number is always irrational.
Proof: Let the rational number be and the irrational number be . We will prove this statement by contradiction. Suppose that their sum is rational, of the form , then we know that , or that , which is rational. This contradicts the condition that is irrational. Hence the sum is always irrational.
Can you answer the following:
A) What do we know about the product of 2 rational numbers? Is it always rational?
B) What do we know about the sum of 2 irrational numbers? Is it always irrational?
C) What do we know about the product of a rational number and an irrational number? Is it always irrational? [Hint: Be very careful!]
Can someone give me feedback? Is this too hard for Cosines group, or just right? Do you want to see more basic material?