Is the Sum of Two Periodic Functions Always Periodic?

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The sum of two periodic functions is generally periodic if their periods have a least common multiple. However, if the least common multiple does not exist, as in the case of sin(x) and sin(pi*x), the sum may not be periodic. Specifically, for the sum to be periodic, the ratio of their periods must be rational. This means that if the periods are incommensurable, the resulting function may not exhibit periodic behavior. Therefore, the periodicity of the sum depends on the relationship between the periods of the individual functions.
Cemre
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Hello,

correct me if I am wrong, but as far as I know if 2 functions f(x) and g(x) are periodic with Tf and Tg periods. f(x)+g(x) is also periodic with least common multiple of Tf and Tg.

But; what if that least common multiple doesn't exist?

is "sin(x) + sin(pi*x)" periodic?

there is no x ( except for zero ) which makes both sin(x) and sin(pi*x) zero at the same x.

Regards.
 
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The quotient of the periods, Tf/Tg has to be rational in order for the sum to be periodic.
 

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