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Scientific Notation

Writing and reading scientific notation

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Scientific Notation
License: CC BY-NC 3.0

Brian is doing research on the Sun for science class. He finds an article that states that the average distance from Earth to the Sun is \begin{align*}1.496 \times 10^8\end{align*}1.496×108 kilometers. Brian is confused because 1.496 doesn't seem like a very big number, and he knows that Earth is far away from the Sun. How can Brian correctly interpret the distance from Earth to the Sun that he found in his research?

In this concept, you will learn how to write very large and very small numbers in scientific notation.

Writing Numbers in Scientific Notation

Very large and very small numbers are used frequently in science. Here are some examples.

  • The distance between Earth and Jupiter is about 595,000,000 kilometers.
  • The diameter of an insect's cell is about 0.000000000017 meters.

To make it easier to read, write, and calculate these extreme numbers, scientists use scientific notation. Scientific notation is a way of representing a very large or very small number without having to write all of the zeros at the beginning or end of the number.

When a number is written in scientific notation it is written as a product of a number that is at least 1 but less than 10 multiplied by a power of 10. Large numbers (numbers greater than 1) are written with a positive power of ten. Small numbers (numbers between 0 and 1) are written with a negative power of ten. The specific power of 10 indicates just how big or how small the number is.

Here are the same quantities from before written in scientific notation. 

  • \begin{align*}595, \!000, \!000=5.95 \times 10^8\end{align*}595,000,000=5.95×108 
  • \begin{align*}0.000000000017=1.7 \times 10^{\text{-}11}\end{align*}0.000000000017=1.7×10-11

Notice that the first number is very large and it has a positive exponent on the 10. The second number is very small and it has a negative exponent on the 10. Also notice that when written in scientific notation, both numbers are the product of a decimal number less than 10 and a power of 10.

Here are the steps for writing a number in scientific notation.

  1. Move the decimal point so that it is to the right of the first non-zero digit of the number. The result should be a number that is at least 1 but less than 10. This will be the first part of your number in scientific notation.
  2. Count how many spaces you needed to move your decimal point in step 1. The number of spaces will be your power of 10. If you moved the decimal point to the left, your exponent will be positive. If you moved your decimal point to the right, your exponent will be negative.
  3. The number in scientific notation is the decimal number from step 1 multiplied by 10 to the power from step 2.

Here is an example.

Write 595,000,000 in scientific notation.

Start by finding the first non-zero digit and put a decimal point to its right. Here, the 5 at the beginning of the number is the first non-zero digit.

5.95000000 which is equal to 5.95

Notice that you don't need to write the zeros at the end of the number anymore because they are to the right of a decimal point.

Next, count how many spaces you needed to move the decimal point to get from 595,000,000 to 5.95. Remember that in 595,000,000 the decimal point is at the very end.

595,000,000 going to 5.95: Move the decimal point 8 spaces to the left.

Now, put everything together. Your number in scientific notation is 5.95 multiplied by 10 to the power of 8. Because you have a very large number and you moved the decimal point to the left in the first step, your exponent will be positive.

\begin{align*}5.95\times 10^8\end{align*}5.95×108

The answer is \begin{align*}595, \!000, \!000=5.95\times 10^8.\end{align*}595,000,000=5.95×108.

Sometimes you will be given a number in scientific notation and you will want to write it as a regular number not in scientific notation. To do this, just follow the steps in reverse.

Here is an example.

Write \begin{align*}3.24\times 10^{\text{-}5}\end{align*}3.24×10-5 as a number not in scientific notation.

First, look at the exponent on the 10. The exponent is -5. Because the exponent is negative, your number is a very small number less than 1 and you will be moving the decimal point to the left to get back to the original number.

Next, move the decimal point on the 3.24. You will move the decimal point 5 spaces to the left. Insert zeros into any blank spaces.

\begin{align*}3.24 \ \text{goes to} \ 0.0000324\end{align*}3.24 goes to 0.0000324

Notice that the result is a small number less than 1. This is exactly what you wanted since the number in scientific notation had a negative exponent.

The answer is \begin{align*}3.24\times 10^{\text{-}5}=0.0000324.\end{align*}3.24×10-5=0.0000324.

Examples

Example 1

Earlier, you were given a problem about Brian and his research on the Sun.

He found that the average distance from Earth to the Sun is \begin{align*}1.496\times 10^8\end{align*}1.496×108 kilometers and he wasn't sure what that meant.

First, Brian needs to realize that the distance is given in scientific notation. The distance isn't 1.496 kilometers, it's a lot more than that!

To write the distance not in scientific notation Brian should look at the exponent on the 10. The exponent is 8. Because the exponent is positive, he will be moving the decimal point to the right to get back to the original number.

Next, he should move the decimal point on the 1.496. He needs to move the decimal point 8 spaces to the right. He should insert zeros into any blank spaces.

\begin{align*}1.496 \ \text{goes to} \ 149,600,000\end{align*}1.496 goes to 149,600,000

Notice that the result is a large number. This makes sense because Brian knows that Earth and the Sun are far apart.

The answer is the average distance from Earth to the Sun is 149,600,000 kilometers.

Example 2

Write \begin{align*}4.5\times 10^{\text{-}6}\end{align*}4.5×10-6 as a number not in scientific notation.

First, look at the exponent on the 10. The exponent is -6. Because the exponent is negative, your number is a small number less than 1 and you will be moving the decimal point to the left to get back to the original number.

Next, move the decimal point on the 4.5. You will move the decimal point 6 spaces to the left. Insert zeros into any blank spaces.

\begin{align*}4.5 \ \text{goes to} \ 0.0000045\end{align*}4.5 goes to 0.0000045

Notice that the result is a small number less than 1. This is exactly what you wanted since the number in scientific notation had a negative exponent.

The answer is \begin{align*}4.5\times 10^{\text{-}6}=0.0000045.\end{align*}4.5×10-6=0.0000045.

Example 3

Write 450,000,000 in scientific notation.

Start by finding the first non-zero digit and put a decimal point to its right. Here, the 4 at the beginning of the number is the first non-zero digit.

4.50000000 which is equal to 4.5

Next, count how many spaces you needed to move the decimal point to get from 450,000,000 to 4.5.

450,000,000. going to 4.5: Move the decimal point 8 spaces to the left.

Now, put everything together. Your number in scientific notation is 4.5 multiplied by 10 to the power of 8. Because you have a very large number and you moved the decimal point to the left in the first step, your exponent will be positive.

\begin{align*}4.5\times 10^8\end{align*}4.5×108

The answer is \begin{align*}450, \!000, \!000=4.5\times 10^8.\end{align*}450,000,000=4.5×108.

Example 4

Write \begin{align*}3.4\times 10^5\end{align*}3.4×105 as a number not in scientific notation.

First, look at the exponent on the 10. The exponent is 5. Because the exponent is positive, your number is a large number greater than 1 and you will be moving the decimal point to the right to get back to the original number.

Next, move the decimal point on the 3.4. You will move the decimal point 5 spaces to the right. Insert zeros into any blank spaces.

\begin{align*}3.4 \ \text{goes to} \ 340,000\end{align*}3.4 goes to 340,000

Notice that the result is a large number. This is exactly what you wanted since the number in scientific notation had a positive exponent.

The answer is \begin{align*}3.4\times 10^5=340,000.\end{align*}3.4×105=340,000.

Example 5

Write 0.0000000067 in scientific notation.

Start by finding the first non-zero digit and put a decimal point to its right. Here, the 6 is the first non-zero digit.

\begin{align*}6.7\end{align*}6.7

Next, count how many spaces you needed to move the decimal point to get from 0.0000000067 to 6.7.

0.0000000067 going to 6.7: Move the decimal point 9 spaces to the right.

Now, put everything together. Your number in scientific notation is 6.7 multiplied by 10 to the power of -9. Because you have a very small number and you moved the decimal point to the right in the first step, your exponent will be negative.

\begin{align*}6.7\times 10^{\text{-}9}\end{align*}6.7×10-9

The answer is \begin{align*}0.0000000067=6.7\times 10^{\text{-}9}.\end{align*}0.0000000067=6.7×10-9.

Review

Write each number in scientific notation.

  1. 0.0000000056731
  2. 24,010,000,000
  3. 960,000,000,000,000,000
  4. 0.0000001245
  5. 36,000,000
  6. 0.00098
  7. 0.000000034
  8. 345,000,000

Write each number as a number not in scientific notation.

  1. \begin{align*}3.808\times10^{11}\end{align*}3.808×1011 
  2. \begin{align*}2.1\times10^6\end{align*}2.1×106 
  3. \begin{align*}5.912\times10^8\end{align*}5.912×108 
  4. \begin{align*}6.78\times10^{-6}\end{align*}6.78×106 
  5. \begin{align*}5.7\times10^9\end{align*}5.7×109 
  6. \begin{align*}4.5\times10^{-5}\end{align*}4.5×105 
  7. \begin{align*}3.21\times10^7\end{align*}3.21×107 

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.16.

Resources

 

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Vocabulary

Exponential Form

The exponential form of an expression is b^x=a, where b is the base and x is the exponent.

order of magnitude

Formally, the order of magnitude is the exponent in scientific notation. Informally it refers to size. Two objects or numbers are of the same order of magnitude are relatively similar sizes.

Scientific Notation

Scientific notation is a means of representing a number as a product of a number that is at least 1 but less than 10 and a power of 10.

Standard Form

As opposed to scientific notation, standard form means writing numbers in the usual way with all of the zeros accounted for in the value.

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