What will happen if I multiply a ket vector by a complex number? I was reading Zettili’s Quantum Mechanics book. There I have seen

when a ket (or bra) multiplied by complex number, we also get a ket (or bra)

But how do we infer this by mathematics?
 A: The states of a quantum system, the kets, are elements of a complex Hilbert space (modulo a phase). A complex Hilbert space is nothing more than a fancy (in)finite dimensional vector space equipped with an inner product. So being the Hilbert space a vector space, all the rules which define a vector space apply. The field upon which the vector space is constructed is the one of complex numbers and so to every vector $|\psi\rangle\in\mathcal{H}$ there is another vector $c|\psi\rangle\in\mathcal{H}$ where $c\in\mathbb{C}$ since in vector spaces we can multiply any element by an element of the underlying field, which in this case is the complex numbers.
A: Think about this like that,- when you multiply a vector by a scalar, you just re-scale original vector. Same idea applies here to vectors in Hilbert space, aka. ket. You just re-scale Hilbert vector by a complex number, an hence get another ket in same Hilbert space.
A: Two things happen, depending on the magnitude and phase of the complex factor. Both can be seen by examining what happens when you insert the factor $f$ in $\mid\Psi\rangle = f\mid\psi\rangle$.
Your primary observables are always bra-ket pairs, so calculate $\langle\Psi\mid\Psi\rangle=\langle\psi\mid f^*f\mid\psi\rangle = |f|^2\langle\psi\mid\psi\rangle$. ($f^*$ is applied to the bra, since it's the complex conjugate of the ket.) That is, multiplying a state by a complex value multiplies the number of particles by the square of the $f$'s magnitude. The phase of $f$ can only be observed via interference with the original $\mid\psi\rangle$, since phase is generally unobservable.
Since operators like mass and momentum are linear, increasing the number increases mass & momentum proportionately. If there are self-interactions in the energy or other operators, those can change in non-linear fashion.
