# How is 0.00124 expressed in proper scientific notation?

Aug 11, 2016

$1.24 \cdot {10}^{- 3}$

#### Explanation:

A number expressed in scientific notation must have the form

$\textcolor{w h i t e}{a a} \textcolor{b l u e}{m} \times {10}^{\textcolor{p u r p \le}{n} \textcolor{w h i t e}{a} \stackrel{\textcolor{w h i t e}{a a a a a a}}{\leftarrow}} \textcolor{w h i t e}{a \textcolor{b l a c k}{\text{the")acolor(purple)("exponent}} a a}$
$\textcolor{w h i t e}{\frac{a}{a} \textcolor{b l a c k}{\uparrow} a a a a}$
$\textcolor{w h i t e}{\textcolor{b l a c k}{\text{the")acolor(blue)("mantissa}} a}$

For normalized scientific notation, the mantissa must satisfy the condition

$1 \le | m | < 10$

Your starting number can be written as

$\textcolor{b l u e}{0.00124} \cdot {10}^{\textcolor{p u r p \le}{0}}$

Your goal here will be to move the decimal place to the right until the mantissa satisfies the above condition. For every position that you move the decimal to the right you must subtract $1$ from the value of the exponent, which starts at $0$.

So, start moving the decimal place to the right

$\textcolor{b l u e}{0.0124} \cdot {10}^{\textcolor{p u r p \le}{- 1}} \to$ one position

$\textcolor{w h i t e}{0} \textcolor{b l u e}{0.124} \cdot {10}^{\textcolor{p u r p \le}{- 2}} \to$ two positions

$\textcolor{w h i t e}{00} \textcolor{b l u e}{1.24} \cdot {10}^{\textcolor{p u r p \le}{- 3}} \to$ three positions

At this point, the mantissa satisfies the required condition

$1 \le \textcolor{b l u e}{1.24} < 1$

which means that you have found the scientific notation for your original number

$0.00124 = 1.24 \cdot {10}^{- 3}$