# 2.17: Comparison of Numbers in Scientific Notation

Difficulty Level: At Grade Created by: CK-12
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Practice Comparison of Numbers in Scientific Notation

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Michelle’s brother is taking chemistry this year and is currently learning about electrons and protons. He has learned that electrons are particles with a negative charge while protons are particles with a positive charge. He has also learned that the mass of an electron is about \begin{align*}9.1094 \times 10^{-31} \end{align*} kilograms while the mass of a proton is about \begin{align*}1.6726 \times 10^{-27}\end{align*} kilograms. At first he thinks that electrons have a greater mass than protons because 9.1094 is greater than 1.6726, but now he’s not sure. How can Michelle use her knowledge of scientific notation to help her brother to compare the mass of an electron with the mass of a proton?

In this concept, you will learn how to compare and order numbers in scientific notation.

### Comparing Numbers Written in Scientific Notation

Recall that 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 and 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.

You can compare two numbers written in scientific notation by looking at their powers of 10. The number with the greater power of 10 will be the greater number. If two numbers have the same power of 10, then compare the decimal numbers to determine the greater number.

Here is an example.

Compare \begin{align*}8.43 \times 10^{6}\end{align*} and \begin{align*}2.38 \times 10^{8}\end{align*}.

First, notice the exponents are 6 and 8. Because the exponents are different, you know that the number with the greater exponent is the greater number. 8 is greater than 6.

\begin{align*}2.38 \times 10^{8}\end{align*} is the greater number.

The answer is \begin{align*}8.45 \times 10^{6} < 2.38 \times 10^{8}\end{align*}.

Here is another example.

Compare \begin{align*}3.2 \times 10^{-10}\end{align*} and \begin{align*}1.2 \times 10^{-9}\end{align*}.

First, notice the exponents are -10 and -9. Even though the exponents are negative, because they are different you still know that the number with the greater exponent is the greater number. -9 is greater than -10.

\begin{align*}1.2 \times 10^{-9}\end{align*} is the greater number.

The answer is \begin{align*}3.2 \times 10^{-10} < 1.2 \times 10^{-9}\end{align*}.

Here is one more example.

Compare \begin{align*}5.65 \times 10^{5}\end{align*} and \begin{align*}5.56 \times 10^{5}\end{align*}.

First, notice the exponents are 5 and 5. Because the exponents are the same, you will have to compare the decimal numbers to determine the greater number. 5.65 is greater than 5.56.

\begin{align*}5.65 \times 10^{5}\end{align*} is the greater number.

The answer is \begin{align*}5.65 \times 10^{5} > 5.56 \times 10^{5}\end{align*}.

Sometimes you will want to compare two numbers, but only one of them is in scientific notation. In this case, convert the number that is not in scientific notation into scientific notation first. Then, compare the two numbers.

### Examples

#### Example 1

Earlier, you were given a problem about Michelle’s brother, who is learning about electrons and protons.

He wants to know how the mass of an electron compares to the mass of a proton. He knows that the mass of an electron is about \begin{align*}9.1094 \times 10^{-31}\end{align*} kilograms while the mass of a proton is about \begin{align*}1.6726 \times 10^{-27}\end{align*} kilograms.

First, Michelle should notice that the numbers are in scientific notation and the exponents are -31 and -27. The exponents are different, so the number with the greater exponent is the greater number. -27 is greater than -31.

\begin{align*}1.6726 \times 10^{-27}\end{align*} is the greater number.

The answer is \begin{align*}9.1094 \times 10^{-31} < 1.6726 \times 10^{-27}\end{align*}. The mass of an electron is smaller than the mass of a proton. In fact, it is almost 2000 times smaller!

#### Example 2

Compare \begin{align*}3.4 \times 10^{5}\end{align*} and \begin{align*}34,000,000\end{align*}.

First, notice that the second number is not in scientific notation. Rewrite 34,000,000 in scientific notation. Find the first non-zero digit and put a decimal point to its right. Here, the 3 is the first non-zero digit.

\begin{align*}3.4\end{align*}

Next, count how many spaces you needed to move the decimal point to get from 34,000,000 to 3.4.

34,000,000 going to 3.4: Move the decimal point 7 spaces to the left.

Now, put everything together. Your number in scientific notation is 3.4 multiplied by 10 to the power of 7.

\begin{align*}3.4 \times 10^{7}\end{align*}

Next, compare the numbers.

\begin{align*}3.4 \times 10^{5}\end{align*} and \begin{align*}3.4 \times 10^{7}\end{align*}

Notice the exponents are 5 and 7. Because the exponents are different, you know that the number with the greater exponent is the greater number. 7 is greater than 5.

\begin{align*}3.4 \times 10^7\end{align*} is the greater number.

The answer is \begin{align*}3.4 \times 10^{5} < 3.4 \times 10^{7}\end{align*} which means \begin{align*}3.4 \times 10^{5} < 34,000,000\end{align*}.

#### Example 3

Compare \begin{align*}4.5 \times 10^{7}\end{align*} and \begin{align*}4.5 \times 10^{9}\end{align*}.

First, notice the exponents are 7 and 9. The number with the greater exponent is the greater number. 9 is greater than 7.

\begin{align*}4.5 \times 10^{9}\end{align*} is the greater number.

The answer is \begin{align*}4.5 \times 10^{7} < 4.5 \times 10^{9}\end{align*}.

#### Example 4

Compare \begin{align*}5.6 \times 10^{-3}\end{align*} and \begin{align*}7.8 \times 10^{-5}\end{align*}.

First, notice the exponents are -3 and -5. The number with the greater exponent is the greater number. -3 is greater than -5.

\begin{align*}5.6 \times 10^{-3}\end{align*} is the greater number.

The answer is \begin{align*}5.6 \times 10^{-3} > 7.8 \times 10^{-5}\end{align*}.

#### Example 5

Compare \begin{align*}8.9 \times 10^{2}\end{align*} and \begin{align*}9.8 \times 10^{2}\end{align*}.

First, notice the exponents are 2 and 2. Because the exponents are the same, you will have to compare the decimal numbers to determine the greater number. 9.8 is greater than 8.9.

\begin{align*}9.8 \times 10^{2}\end{align*} is the greater number.

The answer is \begin{align*}8.9 \times 10^{2} < 9.8 \times 10^{2}\end{align*}.

### Review

Compare the following. Write \begin{align*}< \end{align*}, \begin{align*}>\end{align*}, or \begin{align*}=\end{align*} for each blank.

1. \begin{align*}2.1\times10^6 \ \underline{\;\;\;\;\;\;\;\;} \ 8.9\times10^5\end{align*}
2. \begin{align*}0.00000212 \ \underline{\;\;\;\;\;\;\;\;} \ 2.12\times10^{-5}\end{align*}
3. \begin{align*}4.26\times10^{10} \ \underline{\;\;\;\;\;\;\;\;} \ 4,260,000,000\end{align*}
4. \begin{align*}7.2\times10^{-3} \ \underline{\;\;\;\;\;\;\;\;} \ 1.25\times10^{-1}\end{align*}
5. \begin{align*}2.1 \times 10^{5} \ \underline{\;\;\;\;\;\;\;\;} \ 3.1 \times 10^{4}\end{align*}
6. \begin{align*}8.1 \times 10^{6} \ \underline{\;\;\;\;\;\;\;\;} \ 8.9 \times 10^{5}\end{align*}
7. \begin{align*}7.2 \times 10^{-4} \ \underline{\;\;\;\;\;\;\;\;} \ 8.9 \times 10^{-5}\end{align*}
8. \begin{align*}3.3 \times 10^{2} \ \underline{\;\;\;\;\;\;\;\;} \ 3.1 \times 10^{3}\end{align*}
9. \begin{align*}1.21 \times 10^{4} \ \underline{\;\;\;\;\;\;\;\;} \ 1.89 \times 10^{2}\end{align*}
10. \begin{align*}5.5 \times 10^{6} \ \underline{\;\;\;\;\;\;\;\;} \ 5.51 \times 10^{6}\end{align*}
11. \begin{align*}1.61 \times 10^{3} \ \underline{\;\;\;\;\;\;\;\;} \ 1.89 \times 10^{2}\end{align*}

Order the following from greatest to least. Where necessary, convert to scientific notation.

1. \begin{align*}9.2 \times 10^{10}, 6.4 \times 10^{15} , 2.1 \times 10^{20}, 1.7 \times 10^{15} \end{align*}
2. \begin{align*}5.63 \times 10^{-5}, 4.16 \times 10^{-3} , 3.42 \times 10^{-6}, 8.71 \times 10^{-3} \end{align*}
3. \begin{align*}2.21 \times 10^{-5}, 0.0000202, 2.02 \times 10^{-4}, 0.0000221 \end{align*}
4. \begin{align*}86,102,000, 8.61 \times 10^{8},86,120,000, 8.61 \times 10^{7}\end{align*}

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### Vocabulary Language: English

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