# At the start of 2012, the population of the United States was approximately 313,269,000. What is this number written in scientific notation?

Jun 12, 2016

$313269000 = 3.13269 \times {10}^{8}$

#### Explanation:

In scientific notation, we write a number so that it has single digit to the left of decimal sign and is multiplied by an integer power of $10$.

Note that moving decimal $p$ digits to right is equivalent to multiplying by ${10}^{p}$ and moving decimal $q$ digits to left is equivalent to dividing by ${10}^{q}$.

Hence, we should either divide the number by ${10}^{p}$ i.e. multiply by ${10}^{- p}$ (if moving decimal to right) or multiply the number by ${10}^{q}$ (if moving decimal to left).

In other words, it is written as $a \times {10}^{n}$, where $1 \le a < 10$ and $n$ is an integer.

To write $313269000$ in scientific notation, we will have to move the decimal point eight point to the left, which literally means dividing by ${10}^{8}$ for which we should compensate by multiplying it by ${10}^{8}$.

Hence in scientific notation $313269000 = 3.13269 \times {10}^{8}$.

Jun 14, 2016

$3.13269000 \times {10}^{8}$

#### Explanation:

Given:$\text{ } 313269000$

Objective: have a single non zero digit to the left of the decimal point. In doing this you have changed the value so a mathematical correction will need to be applied.

Multiply by 1 but in the form of $1 = {10}^{8} / {10}^{8}$ giving

$313269000 \times {10}^{8} / {10}^{8} \text{ "larr" this does not change the value}$

$\frac{313269000}{10} ^ 8 \times {10}^{8}$

$3.13269000 \times {10}^{8} \text{ "larr" same value; just looks different!}$

The ${10}^{8}$ is the mathematical correction which is in place but not applied.