Did you know that the average distance of the Earth from the Sun is about 92,000,000 miles? This is a big number! Do you think that there is any way to write it more compactly?

### Scientific Notation

Sometimes in mathematics numbers are huge. They are so huge that we use what is called scientific notation. It is easier to work with such numbers when we shorten their decimal places and multiply them by 10 to a specific power.

A number is expressed in **scientific notation** when it is in the form:

\begin{align*}N \times 10^n\end{align*}

where \begin{align*}1\le N <10\end{align*} and \begin{align*} n \end{align*} is an integer.

For example, \begin{align*} 2.35 \times 10^{37}\end{align*} is a number expressed in scientific notation. Notice there is only one number in front of the decimal place.

Since the scientific notation uses powers of ten, we want to be comfortable expressing different powers of ten.

Powers of 10:

\begin{align*}100,000 &= 10^5\\ 10,000 &= 10^4\\ 1,000 &= 10^3\\ 100 &= 10^2\\ 10 &= 10^1\end{align*}

**Using Scientific Notation for Large Numbers**

If we divide 643,297 by 100,000 we get 6.43297. If we multiply 6.43297 by 100,000, we get back to our original number, 643,297. But we have just seen that 100,000 is the same as \begin{align*}10^5\end{align*}, so if we multiply 6.43297 by \begin{align*}10^5\end{align*}, we should also get our original number, 643,297, as well. In other words \begin{align*}6.43297 \times 10^5=643,297\end{align*}. Because there are five zeros, the decimal moves over five places.

Look at the following examples:

\begin{align*}2.08 \times 10^4 &= 20,800\\ 2.08 \times 10^3 &= 2,080\\ 2.08 \times 10^2 &= 208\\ 2.08 \times 10^1 &= 20.8\\ 2.08 \times 10^0 &= 2.08\end{align*}

The power tells how many decimal places to move; positive powers mean the decimal moves to the right. A positive 4 means the decimal moves four positions to the right.

#### Let's write 653,937,000 in scientific notation:

\begin{align*}653,937,000=6.53937000 \times 100,000,000=6.53937 \times 10^8\end{align*}

Oftentimes, we do not keep more than a few decimal places when using scientific notation, and we round the number to the nearest whole number, tenth, or hundredth depending on what the directions say. Rounding Example A could look like \begin{align*}6.5 \times 10^8\end{align*}.

#### Using Scientific Notation for Small Numbers

We’ve seen that scientific notation is useful when dealing with large numbers. It is also good to use when dealing with extremely small numbers.

Look at the following examples:

\begin{align*}2.08 \times 10^{-1} &= 0.208\\ 2.08 \times 10^{-2} &= 0.0208\\ 2.08 \times 10^{-3} &= 0.00208\\ 2.08 \times 10^{-4} &= 0.000208\end{align*}

#### Let's write 0.000539 in scientific notation:

\begin{align*}0.000539=5.39 \times 0.0001=5.39 \times 10^{-4}\end{align*}

### Examples

#### Example 1

Earlier, you were asked how to write the distance of the Earth from the Sun, 92,000,000 miles, in a more compact way.

You can write this number in scientific notation. In order to get a number between 1 and 10, you need to move the decimal point 7 places to the left.

\begin{align*}92,000,000&=9.2\times 10000000\\ &=9.2\times 10^7\end{align*}

#### Example 2

The time taken for a light beam to cross a football pitch is 0.0000004 seconds. Write in scientific notation.

\begin{align*}0.0000004=4 \times 0.0000001=4 \times \frac{1}{10,000,000}=4 \times \frac{1}{10^7}=4 \times 10^{-7}\end{align*}

### Review

Write the numerical value of the following expressions.

- \begin{align*}3.102 \times 10^2\end{align*}
- \begin{align*}7.4 \times 10^4\end{align*}
- \begin{align*}1.75 \times 10^{-3}\end{align*}
- \begin{align*}2.9 \times 10^{-5}\end{align*}
- \begin{align*}9.99 \times 10^{-9}\end{align*}

Write the following numbers in scientific notation.

- 120,000
- 1,765,244
- 63
- 9,654
- 653,937,000
- 1,000,000,006
- 12
- 0.00281
- 0.000000027
- 0.003
- 0.000056
- 0.00005007
- 0.00000000000954

#### Quick Quiz

- Simplify: \begin{align*}\frac{(2x^{-4}y^3)^{-3} \ \cdot \ x^{-3} y^{-2}}{-2x^0y^2}\end{align*}.
- The formula \begin{align*}A=1,500(1.0025)^t\end{align*} gives the total amount of money in a bank account with a balance of $1,500.00, earning 0.25% interest, compounded annually. How much money would be in the account five years in the past?
- True or false?\begin{align*}\left(\frac{5}{4}\right)^{-3}= -\frac{125}{64}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.5.