Can you determine if -1.3 is rational, irrational, or not a real number.?
rational numbers have either a discrete (not infinite) number of decimal places (this number can be zero) or an infinite non repeating value such as pi or the natural logarithm e irrational numbers are all the rest non-real numbers (also called complex numbers) are numbers that are not defined in normal mathematics such as the square root of negative one in specific answer -1.3 is rational because it has a discrete number of desimal places
How to find out if this answer is irrational?
This can be done as a proof by contradiction (Reductio ad absurdum). Start by assuming E is rational, meaning it can be expressed as the reduced fraction p/q with p and q both positive integers, no common factors between them (p and q are co-prime). Then just play with the algebra. 14^(p/q) = 9 14^p = 9^q Prime-factorise: (2^p)(7^p) = 3^(2q) Both LHS and RHS are integers because p and q are both positive integers by the initial assumption, and this equation states that they are equal. Now, the LHS is divisible by 2 and 7 while the RHS is not, and the RHS is divisible by 3, but the LHS is not (this is due to the uniqueness of prime factorisation). So the LHS equals the RHS but they are also co-prime. This is an absurdity. So the original assumption must be false and E is irrational.
For what value(s) of x is f continuous?
This function is called "Dirichlet's function." It is a well-known function that is discontinuous everywhere. Therefore, the answer is D) None. I hope this helps!