Period of sum of periodic functions

Question

Why is the following function:

$$f(t) = \sin(2\pi\cdot10^6t) + \sin(2\pi\cdot10^6t) \sin(2\pi\cdot1000t)$$

periodic, whereas

$$f(t) = 10\cdot \sin(2\pi\cdot10^6t) + 10\cdot 0.5\cdot \sin(2\pi\cdot10^6t) \sin(2\pi\cdot1000t)$$

is not periodic?

Answer 1

Mathematica (which is the engine running under WolframAlpha) treats any decimal values such as 0.5 as imprecise decimal approximations. As such it reports there is no period as it is not treating 0.5 as exactly equal to $\frac{1}{2}$. If you replace the $0.5$ with $1/2$ in the input to WolframAlpha it will give you that the period is $\frac{1}{1000}$.