# Is cosx+cos(sqrt(2)x) periodic?

1. Nov 24, 2012

### peripatein

1. The problem statement, all variables and given/known data

Is the function cosx + cos(sqrt(2)x) is periodic?

2. Relevant equations

cos(x)=cos(x+2pi)

3. The attempt at a solution

For the above function to be periodic:
cosx + cos(sqrt(2)x) = cos(x+T) + cos(sqrt(2)(x + T))
Does that imply that 2pi = T AND 2pi = sqrt(2)T, ergo there is no such T?

2. Nov 24, 2012

### Millennial

It doesn't exactly imply that. However, by the definition of periodicity, you need integer T to satisfy the equation. For example, the function cos(2x)+cos(3x) is periodic, one has period $\pi$ and the other has period $2\pi/3$. It is not hard to see that both of these functions are also periodic by $2\pi$, and hence their sum is also periodic. Returning to your question, your answer is correct, the function is not periodic. However, your equations seem a bit fallacious to me. All you need is that the period of the sum is the least common multiple of the periods of the summands.

Can you see such a least common multiple?