Weirdness when converting in scientific notation?

[foxaquinn]

I apologize if this isn't the right place for this!

Why is it, when converting this:
$$373 * 10^-9$$to this:
$$3.73*10^-7$$the10^-9 becomes 10^-7 and not 10^-11? It seems like since you're dividing 373 by 100 in order to move the decimal point, you'd divide 10^-9 by 100 as well, but instead, it seems to multiply by 100.

 

[Steven G]

That one is easy to answer. If you divide one number by 100 and then another number by 100, then you are dividing the product of the two numbers by 10,000! That will not be the same result.

For example does $\displaystyle 5*7 = \dfrac {5}{100}\dfrac{7}{100} = \dfrac{35}{10000}?$

Does $\displaystyle 5*7 = (\dfrac {5}{100})(7*100) ?$

 

[Otis]

$\displaystyle 373 * 10^-9$
to this:
$\displaystyle 3.73*10^-7$
the10^-9 becomes 10^-7 and not 10^-11?

Hello foxaquinn. Here's another way to see it.

$\displaystyle 373×10^{-9} = \frac{373}{10^9}$

Divide numerator and denominator by 100.

$\displaystyle \frac{3.73}{\frac{10^9}{10^2}}$

We simplify $\frac{10^9}{10^2}$ by exponent property: $10^{9-2}=10^7$

$\displaystyle \frac{3.73}{10^7} = 3.73×10^{-7}$

Yet another check on our thinking: We know that the exponent in scientific notation tells us how many places to shift the decimal point (with the exponent's sign telling us which direction).

373.0 × 10^(-9) means the decimal point shifts nine positions to the left.

When we write 3.73, the decimal point has already shifted two positions, so now it only needs to go another seven positions (to reach the goal of nine places from where it started).

3.73 × 10^(-7)

?

$\;$