Multiplication by Power of 10 by Moving Decimal Point

Theorem
Let $n \in \R$ be a real number.

Let $n$ be expressed in decimal notation.

Let $10^d$ denote a power of $10$ for some integer $d$

Then $n \times 10^d$ can be expressed in decimal notation by shifting the decimal point $d$ places to the right.

Thus, if $d$ is negative, and so $10^d = 10^{-e}$ for some $e \in \Z_{>0}$, $n \times 10^d$ can be expressed in decimal notation by shifting the decimal point $e$ places to the left.

Proof
Let $n$ be expressed in decimal notation as:


 * $n = \sqbrk {a_r a_{r - 1} \dotso a_1 a_0 \cdotp a_{-1} a_{-2} \dotso a_{-s} a_{-s - 1} \dotso}$

That is:
 * $n = \ds \sum_{k \mathop \in \Z} a_k 10^k$

Then:

The effect of presenting digit $a_{k - d}$ in position $k$ of $n$ is the same as what you get having moved the decimal point from between $a_0$ and $a_{-1}$ to between $a_{-d}$ and $a_{-d - 1}$.

Thus:
 * if $d$ is positive, that is equivalent to moving the decimal point $d$ places to the right

and:
 * if $d$ is negative, that is equivalent to moving the decimal point $d$ places to the left.