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# Comparison of Numbers in Scientific Notation

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Jennifer is writing a paper for English class on different distances in track and field. She has decided to focus her paper on middle and long distances and has made a list of every distance longer than 1 mile. Here is her list.

1,500 m

2000 m

3,000 m

5,000 m

10,000 m

20,000 m

25,000 m

30,000 m

She has also learned about scientific notation in math class. She has decided to write each distance in scientific notation. That way she can add a twist to her paper and surprise both her English and her math teacher.

Think about how Jennifer can accomplish this goal. If you were to rewrite each distance in scientific notation, how would you do it? This Concept will teach you all that you need to know about scientific notation. Let’s get started.

### Guidance

Previously we worked on how to compare and order whole numbers and decimals.

Numbers in scientific notation can be compared and ordered as well. In scientific notation, the number with the greater power of 10 is always the larger number.

Take $9.6 \times 10^3$ compared to $2.2 \times 10^5$ . Let’s look at the numbers in standard form to illustrate the point.

$9.6 \times 10^3 \rightarrow 9,600$ and $2.2 \times 10^5 \rightarrow 220,000$ , therefore, $9.6 \times 10^3 < 2.2 \times 10^5$

Remember to apply what you know about negative numbers to scientific notation with negative powers. When comparing the same number to the powers of $10^{-7}$ and $10^{-11}$ , for example, the number to the power of -7 is the greater value, because $-7 > -11$ .

If the powers of 10 are the same, then we look to the decimals to compare.

Finally, when comparing a number in standard form to a number in scientific notation, convert the number in standard form to scientific notation; then compare.

Compare $8.43 \times 10^6$ and $2.38 \times 10^8$

First, notice that both of the exponents are positive and they are different, so we can simply compare the exponent. The larger the exponent is the larger the number. Here is our answer.

$8.43 \times 10^6 < 2.38 \times 10^8$

Compare $3.2 \times 10^{-10}$ and $1.2 \times 10^{-9}$

First, notice that both of the exponents are negative. Therefore, we have to compare the greater exponent as the larger number. This is a little bit backwards, but remember that negative numbers are larger the closer that they are to zero. Therefore, negative 9 is greater than negative 10.

$3.2 \times 10^{-10} < 1.2 \times 10^{-9}$

Compare $5.65 \times 10^5$ and $5.56 \times 10^5$

First, notice that the exponents are the same here. Therefore, we compare the decimals.

$5.65 \times 10^5 > 5.56 \times 10^5$

When ordering numbers in scientific notation, we do the same work as with comparing. Break the numbers down by looking at the exponents and then write them in order according to the directions.

Now it's time for you to try a few on your own.

#### Example A

$4.5 \times 10^7$ and $4.5 \times 10^9$

Solution: <

#### Example B

$5.6 \times 10^{-3}$ and $7.8 \times 10^{-5}$

Solution: >

#### Example C

$8.9 \times 10^2$ and $9.8 \times 10^2$

Solution:

Here is the original problem once again.

Jennifer is writing a paper for English class on different distances in track and field. She has decided to focus her paper on middle and long distances and has made a list of every distance longer than 1 mile. Here is her list.

1,500 m

2000 m

3,000 m

5,000 m

10,000 m

20,000 m

25,000 m

30,000 m

She has also learned about scientific notation in math class. She has decided to write each distance in scientific notation. That way she can add a twist to her paper and surprise both her English and her math teacher.

To write each distance in scientific notation, Jennifer will need to use powers of 10. Let’s look at the first distance.

1,500 m can be changed to $1.5 \times 10^3$ since we moved the decimal point three places to the left, the exponent is positive.

In fact, we will be moving all of the decimal points to the left in this problem. Here are the other distances written in scientific notation.

$&2 \times 10^3\\&3 \times 10^3\\&5 \times 10^3\\&1 \times 10^4 \ \text{or} \ 10 \times 10^3\\&2 \times 10^4\\&2.5 \times 10^4\\&3 \times 10^4$

Here is our list of answers.

### Vocabulary

Standard Form
the writing of a number with zeros not written using exponents and powers of 10.
Exponential Form
A number written with an exponent
Scientific Notation
Numbers that are written as decimal products with base ten powers

### Guided Practice

Here is one for you to try on your own.

Compare $3.4 \times 10^5$ and $34,000,000$

First, notice that you have one value in scientific notation and one in standard notation.

We should write them both in the same form to make the comparison easier.

Let's write the first value in standard notation.

$3.4 \times 10^5 = 340,000$

Now we compare the two values.

$340,000 < 34,000,000$

### Practice

Directions: Compare the following. Write <, >, or = for each ___.

1. $2.1 \times 10^6 \ \underline{\;\;\;\;\;} \ 8.9 \times 10^5$

2. $0.00000212 \ \underline{\;\;\;\;\;} \ 2.12 \times 10^{-5}$

3. $4.26 \times 10^{10} \ \underline{\;\;\;\;\;} \ 4,260,000,000$

4. $7.2 \times 10^{-3} \ \underline{\;\;\;\;\;} \ 12.5 \times 10^{-1}$

5. $2.1 \times 10^5 \ \underline{\;\;\;\;\;} \ 3.1 \times 10^4$

6. $8.1 \times 10^6 \ \underline{\;\;\;\;\;} \ 8.9 \times 10^5$

7. $7.2 \times 10^(-4) \ \underline{\;\;\;\;\;} \ 8.9 \times 10^(-5)$

8. $3.3 \times 10^2 \ \underline{\;\;\;\;\;} \ 3.1 \times 10^3$

9. $12.1 \times 10^4 \ \underline{\;\;\;\;\;} \ 18.9 \times 10^2$

10. $5.5 \times 10^6 \ \underline{\;\;\;\;\;} \ 5.51 \times 10^6$

11. $16.1 \times 10^3 \ \underline{\;\;\;\;\;} \ 18.9 \times 10^2$

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

12. $9.2 \times 10^{10}, \ 6.4 \times 10^{15}, \ 2.1 \times 10^{20}, \ 1.7 \times 10^{15}$

13. $5.63 \times 10^{-5}, \ 4.16 \times 10^{-3}, \ 3.42 \times 10^{-6}, \ 8.71 \times 10^{-3}$

14. $2.12 \times 10^{-5}, \ 0.0000202, \ 2.02 \times 10^{-4}, \ 0.0000221$

15. $86,102,000, \ 8.61 \times 10^8, \ 86,120,000, \ 8.61 \times 10^7$