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Let $K$ be an algebraically closed field of characteristic different from 2,$g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficientsin $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$hyperelliptic curve over $K$ and $J$ the jacobian of $C$. We identify $C$ withthe image of its canonical embedding into $J$ (the infinite point of $C$ goesto the zero point of $J$). For each point $P=(a,b)\in C(K)$ there are $2^{2g}$points $\frac{1}{2}P \in J(K)$. We describe explicitly the Mumfordrepresesentations of all $\frac{1}{2}P$. The rationality questions for$\frac{1}{2}P$ are also discussed.

Division by 2 on hyperelliptic curves and jacobians