Tate objects $\mathbb{Z}(n)$ in Hodge structures or mixed Hodge structures

Question

The Tate object $\mathbb{Z}(n)$ is defined as the pure Hodge structures with lattice

$$\mathbb{Z}(n)=(2\pi i)^n\mathbb{Z}\subset \mathbb{C}$$

and Hodge filtration induced by

$$[\mathbb{Z}(n)\otimes \mathbb{C}]^{-m,-m}=\mathbb{Z}(m)\otimes \mathbb{C}$$

and zeros in other degree. Why use the lattice $(2\pi i)^n\mathbb{Z}$? Why just use the lattice $\mathbb{Z}$ in the definition?