if xy is rational must either x or y be rational
both must be rational
rational x irritational = irrational
irritational x irritational = irrational
Yes, if the product of two numbers (xy) is rational, then at least one of the numbers (x or y) must be rational.
To understand why, we can consider the definition of rational numbers. Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero. Let's assume that xy is a rational number.
If xy is rational, it means that it can be written as a fraction, such as xy = a/b, where a and b are integers and b is not zero. Now, let's consider the two cases:
Case 1: If x is rational, then we can express it as x = c/d, where c and d are integers and d is not zero. Substituting x = c/d in xy = a/b, we get (c/d)y = a/b. This can be rearranged as y = (b/d)(a/c), which shows that y is also rational since (b/d)(a/c) is a ratio of two integers.
Case 2: If y is rational, then we can express it as y = e/f, where e and f are integers and f is not zero. Substituting y = e/f in xy = a/b, we get x(e/f) = a/b. This can be rearranged as x = (b/f)(a/e), which shows that x is also rational since (b/f)(a/e) is a ratio of two integers.
In both cases, we have shown that if xy is rational, then either x or y (or both) must be rational.