# How should 0.0000497 be expressed in proper scientific notation?

Jul 21, 2016

$4.97 \cdot {10}^{- 5}$

#### Explanation:

For a number written using normalized scientific notation, you have

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

For normalized scientific notation, you must have

1 <= |m| < 10" "color(orange)(("*"))

Your goal here is to find the value of the mantissa and the value of the exponent. To find the mantissa, start to move the decimal point to the right of the given number until you find a value that matches the given condition $\textcolor{\mathmr{and} a n \ge}{\left(\text{*}\right)}$.

For every position you move the decimal point to the right, you must subtract $1$ to the value of the exponent, which starts at $\textcolor{p u r p \le}{n} = 0$.

You will have

$0.0000497 \to \textcolor{p u r p \le}{n} = \textcolor{w h i t e}{-} 0$

$\textcolor{w h i t e}{0} 0.000497 \to \textcolor{p u r p \le}{n} = - 1 \text{ and } 1 \textcolor{red}{\cancel{\textcolor{b l a c k}{\le}}} \textcolor{b l u e}{m} = 0.000497 < 10$

$\textcolor{w h i t e}{00} 0.00497 \to \textcolor{p u r p \le}{n} = - 2 \text{ and } 1 \textcolor{red}{\cancel{\textcolor{b l a c k}{\le}}} \textcolor{b l u e}{m} = \textcolor{w h i t e}{0} 0.00497 < 10$

$\textcolor{w h i t e}{000} 0.0497 \to \textcolor{p u r p \le}{n} = - 3 \text{ and } 1 \textcolor{red}{\cancel{\textcolor{b l a c k}{\le}}} \textcolor{b l u e}{m} = \textcolor{w h i t e}{00} 0.0497 < 10$

$\textcolor{w h i t e}{0000} 0.497 \to \textcolor{p u r p \le}{n} = - 4 \text{ and } 1 \textcolor{red}{\cancel{\textcolor{b l a c k}{\le}}} \textcolor{b l u e}{m} = \textcolor{w h i t e}{000} 0.497 < 10$

color(white)(00000)4.97 -> color(purple)(n)=-5" and " 1 <= color(white)(a)color(blue)(m)=color(white)(0000)4.97 < 10 " "color(darkgreen)(sqrt())

And there you have it. You moved the decimal five positions to the right, which means that the exponent is equal to n = $\textcolor{p u r p \le}{- 5}$. The mantissa is equal to $\textcolor{b l u e}{4.97}$, which means that your number can be written as

$0.0000497 = \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{4.97 \cdot {10}^{- 5}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$