2.6: Scientific Notation
Introduction
The Distances
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
2,000 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 lesson will teach you all that you need to know about scientific notation. Let’s get started.
What You Will Learn
By the end of this lesson you will have an understanding of the following skills:
- Recognize equivalence of standard form, product form and scientific notation of very small and very large decimal numbers.
- Compare and order numbers in scientific notation.
- Find sums, differences, products and quotients of numbers in scientific notation.
- Solve real-world problems involving operations in scientific notation.
Teaching Time
I. Recognize Equivalence of Standard Form, Product Form and Scientific Notation of Very Small and Very Large Decimal Numbers
To better understand our world, scientists take measurements. Some measurements can often be extremely large or small.
The distance between Earth and Jupiter, for instance, is about 595,000,000 kilometers.
The diameter of a single insect cell is about .000000000017 meters.
Scientific notation is a way to represent very large and very small numbers. Scientific notation makes it easier to read, write, and calculate these extreme numbers.
How do we work with scientific notation?
To understand scientific notation, we need to first back up a bit and think about some of the things that we have already learned.
Do you remember working with powers? Exponential notation is a way to write multiplication as a power.
In a power like \begin{align*}2^3\end{align*}
Scientific notation relies on the exponential notation—powers of 10—to rewrite numbers in a simpler form. Let’s look at powers of 10 once again.
Our place-value system is based on these powers of 10.
10(10) = 100 notice that \begin{align*}10^2\end{align*} is a "1" followed by two 0's
100(10) = 1,000 notice here that \begin{align*}10^3\end{align*} is a "1" followed by three 0's
1000(10) = 10,000 and so on.
The pattern continues, so that every time you raise 10 to a power, it equals 1 followed by the number of zeros equal to the number of the power.
Now let’s think about that original distance that we introduced between Earth and Jupiter.
The distance is 595,000,000 miles. This number is written in standard form.
Scientific notation converts a very large (or very small) number into an easier to read number. It does this by moving the decimal place either right or left, depending on the number. In the case of a very large number, scientific notation moves the decimal place left in that number, until all we are left with is a number in the ones place followed by the remaining non-zero numbers after the decimal. Based on what we learned above, we can then convert all the missing decimal places into a 10 with an exponent that represents the number of places we moved the decimal to the left.
Let's look at the number 595,000,000. Because this number does not have any decimal places, a decimal point is not shown. But we could insert a decimal, and it would be at the very end of the number. To convert this number using scientific notation, we would move that decimal place to the left until all we have is a number in the ones place. It would look like this: 5.95000000.
We know that when writing decimals we do not have to show the 0's, so if we remove the zeros we are left with 5.95. This is not our original number; we must remember scientific notation is a way of writing large numbers in a simplified manner. Therefore, we must maintain the true value of the number. We do this by identifying the number of decimal places we moved to the left. In this case it was 8 places. Based on the chart above, we know that we can write \begin{align*}10^8\end{align*} to represent the number 100,000,000. If we took this number and multiplied it by 5.95, we would have our original number of 595,000,000. So we can write the number 595,000,000 as \begin{align*}5.95 x 10^8\end{align*}. That is scientific notation.
Example
Write 560,000 in scientific notation.
To begin with, move the decimal left until we are left with a number in the ones place followed by the remaining numbers.
5.60000 becomes 5.6
Now we can think about how many decimal places to the left we moved. We moved 5 decimal places, so we would write those decimal places as powers of 10.
Our answer is \begin{align*}5.6 \times 10^5\end{align*}
Well, it has to do with the way that the decimal point is moving. If it moves to the left, then the power is positive. If it moves to the right, then the power is negative. In the last example, our decimal moved to the left. Now let’s look at an example where it moves to the right.'
Example
The diameter of the insect’s gland cell is .000000000017.
.000000000017 is the standard form of the number. Notice that that is a very tiny decimal. To write this in scientific notation, we have to move the decimal point 11 places to the right.
.000000000017 becomes 1.7
We moved the decimal 11 places, so that is our exponent.
Now we write the whole thing as a product of the decimal and a power of ten.
Our answer is \begin{align*}1.7 \times 10^{-11}\end{align*}.
We can also work the other way around. We can take a number in scientific notation and rewrite it in standard form. To do this, we have to work backwards. Pay close attention to whether the exponent is negative or positive, and this will tell you which way to move the decimal point.
Example
Write \begin{align*}1.2 \times 10^4\end{align*} in standard form.
First, notice that the exponent is positive. This means that we move the decimal four places to the right.
\begin{align*}1.2 \times 10^4\end{align*} becomes 12,000.
Our answer is 12,000.
Example
Write \begin{align*}4.5 \times 10^{-6}\end{align*} in standard form.
First, notice that the exponent is negative. This means that we move the decimal six places to the left.
\begin{align*}4.5 \times 10^{-6}\end{align*} becomes .0000045.
Our answer is .0000045
2O. Lesson Exercises
- Write 450,000,000 in scientific notation.
- Write \begin{align*}3.4 \times 10^5\end{align*} in standard form.
- Write \begin{align*}6.7 \times 10^{-9}\end{align*} in standard form
Take a few minutes to check your answers with a partner.
Take a few notes on the difference between standard form and scientific notation and then continue with the lesson.
II. Compare and Order Numbers in Scientific Notation
You already know 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 \begin{align*}9.6 \times 10^3\end{align*} compared to \begin{align*}2.2 \times 10^5\end{align*}. Let’s look at the numbers in standard form to illustrate the point.
\begin{align*}9.6 \times 10^3 \rightarrow 9,600\end{align*} and \begin{align*}2.2 \times 10^5 \rightarrow 220,000\end{align*}, therefore, \begin{align*}9.6 \times 10^3 < 2.2 \times 10^5\end{align*}
Remember to apply what you know about negative numbers to scientific notation with negative powers. When comparing the same number to the powers of \begin{align*}10^{-7}\end{align*} and \begin{align*}10^{-11}\end{align*}, for example, the number to the power of -7 is the greater value, because \begin{align*}-7 > -11\end{align*}.
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.
Let’s apply this information by looking at an example.
Example
Compare \begin{align*}8.43 \times 10^6\end{align*} and \begin{align*}2.38 \times 10^8\end{align*}
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.
\begin{align*}8.43 \times 10^6 < 2.38 \times 10^8\end{align*}
Example
Compare \begin{align*}3.2 \times 10^{-10}\end{align*} and \begin{align*}1.2 \times 10^{-9}\end{align*}
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.
\begin{align*}3.2 \times 10^{-10} < 1.2 \times 10^{-9}\end{align*}
Example
Compare \begin{align*}5.65 \times 10^5\end{align*} and \begin{align*}5.56 \times 10^5\end{align*}
First, notice that the exponents are the same here. Therefore, we compare the decimals (.65 is greater than .56.).
\begin{align*}5.65 \times 10^5 > 5.56 \times 10^5\end{align*}
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.
2P. Lesson Exercises
Compare.
- \begin{align*}4.5 \times 10^7\end{align*} and \begin{align*}4.5 \times 10^9\end{align*}
- \begin{align*}5.6 \times 10^{-3}\end{align*} and \begin{align*}7.8 \times 10^{-5}\end{align*}
- \begin{align*}8.9 \times 10^2\end{align*} and \begin{align*}9.8 \times 10^2\end{align*}
Take a few minutes to check your answers with a partner.
III. Find Sums, Differences, Products and Quotients of Numbers in Scientific Notation
Scientific notation makes reading and writing very large and very small numbers easier; it makes computation with such numbers easier as well.
Let’s start with addition and subtraction. Before performing addition or subtraction on scientific notation, the exponents must be the same. Matching the exponents involves a simple case of moving the decimal point—a process you’ve completed many times in making the divisor a whole number before dividing decimals. Let’s see how it’s done in the following addition problem. Note how we use parentheses to group the scientific notation on either side of the addition sign.
\begin{align*}(5.7 \times 10^4) + (4.87 \times 10^5)\end{align*}
We want to make both of these exponents the same. To make both exponents 5’s, we move the decimal point in 5.7 one place to the left by multiplying by 10.
\begin{align*}(.57 \times 10^5) + (4.87 \times 10^5)\end{align*}
Now we can add the decimal parts of the problem. The power of 10 stays the same.
\begin{align*}&(.57 \times 10^5) + (4.87 \times 10^5)\\ &(.57 + 4.87) \times 10^5\\ & 5.44 \times 10^5\end{align*}
Our answer is \begin{align*}5.44 \times 10^5\end{align*}.
Subtraction works the same way as addition: Before performing the subtraction operation, the exponents must be the same.
Multiplication and division in scientific notation is a little different.
Do you remember simplifying exponents?
Example
\begin{align*}(x^3)x^4\end{align*}
To multiply the exponents, we add the powers. \begin{align*}(x^3)x^4 = x^{3 + 4} = x^7\end{align*}. Multiplying scientific notation is similar: You multiply the decimals and add the exponents.
Example
\begin{align*}&(3.4 \times 10^{-2}) \times (6.2 \times 10^6)\\ & (3.4 \times 6.2) \times (10^{-2 + 6})\\ & 21.08 \times 10^4\end{align*}
Division of scientific notation is identical to multiplication—except you divide the decimals and subtract the exponents. Let’s try it out.
Example
\begin{align*}&(8.4 \times 10^5) \div (1.4 \times 10^{-2})\\ & (8.4 \div 1.4) \times (10^{5 - (-2)}) \rightarrow \text{Remember subtracting a negative is the same as adding it.}\\ & 6 \times 10^7\end{align*}
2Q. Lesson Exercises
- Add \begin{align*}(3.4 \times 10^3 + 5.6 \times 10^4)\end{align*}
- Multiply \begin{align*}(1.2 \times 10^6)(3.4 \times 10^4)\end{align*}
- Subtract \begin{align*}(5.6 \times 10^4 - 3.2 \times 10^4)\end{align*}
Check your answers with a friend. Be sure that your work is accurate.
IV. Solve Real-World Problems Involving Operations in Scientific Notation
In our examples of the distance between Earth and Jupiter and the diameter of an insect’s cell, we’ve seen how scientific notation makes such very large and very small numbers more manageable. What other instances can you think of where scientists might get very large or very small measurements? Now that you know how to read, write, and calculate with scientific notation, you can apply your knowledge to real-world scenarios—just like a scientist!
Let’s look at an example before we apply what we have learned to our introductory problem.
Example
At its closest, the planet Neptune is 4,300,000,000 kilometers away from Earth. A group of astronauts from Earth want to make it to Neptune is 20,000 days. If they travel the same amount of kilometers each day, how many kilometers will they travel each day? Convert both numbers to scientific notation before solving.
Let’s begin by converting both numbers to scientific notation.
The distance between Earth and Neptune, in scientific notation, is \begin{align*}4.3 \times 10^9\end{align*}. The number of days the astronauts want to travel in scientific notation is \begin{align*}2.0 \times 10^4\end{align*}.
We want to divide the distance evenly among the days, so we know we need to divide. Remember: To divide numbers in scientific notation, you divide the decimals and subtract the exponents.
\begin{align*}&4.3 \times 10^9 \div 2.0 \times 10^4\\ & (4.3 \div 2.0) \times 10^{9 - 4}\\ & 2.15 \times 10^5\end{align*}
Remember to put the units of measurement in your answer!
Our answer is \begin{align*}2.15 \times 10^5 \ km\end{align*} or 215,000 kilometers.
Real Life Example Completed
The Distances
Here is the original problem once again. Reread it and underline any important information.
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
2,000 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 \begin{align*}1.5 \times 10^3\end{align*} 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.
\begin{align*}&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\end{align*}
If we had a runner who was combining distances, we could also add or subtract these distances to get a new distance. Think back to the lesson and practice adding distances with a partner. Take a few minutes to try a few of these now.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- 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
Technology Integration
Khan Academy Scientific Notation
James Sousa, Scientific Notation
James Sousa, Animation on Scientific Notation
James Sousa, Write a Number in Scientific Notation
James Sousa, Write a Number in Decimal Notation when Given in Scientific Notation
James Sousa, Example of Multiplying Numbers Written in Scientific Notation
James Sousa, Example of Dividing Numbers Written in Scientific Notation
Time to Practice
Directions: Write each number in scientific notation.
1. 0.0000000056731
2. 24,010,000,000
3. 960,000,000,000,000,000
4. 0.0000001245
Directions: Write each number in standard form.
5. \begin{align*}3.808 \times 10^{11}\end{align*}
6. \begin{align*}2.1 \times 10^{-6}\end{align*}
7. \begin{align*}5.912 \times 10^8\end{align*}
8. \begin{align*}1.001 \times 10^{-4}\end{align*}
Directions: Compare the following. Write <, >, or = for each ___.
9. \begin{align*}2.1 \times 10^6 \ \underline{\;\;\;\;\;} \ 8.9 \times 10^5\end{align*}
10. \begin{align*}0.00000212 \ \underline{\;\;\;\;\;} \ 2.12 \times 10^{-5}\end{align*}
11. \begin{align*}4.26 \times 10^{10} \ \underline{\;\;\;\;\;} \ 4,260,000,000\end{align*}
12. \begin{align*}7.2 \times 10^{-3} \ \underline{\;\;\;\;\;} \ 12.5 \times 10^{-1}\end{align*}
Directions: Where necessary, convert to scientific notation. Order the following from greatest to least.
13. \begin{align*}9.2 \times 10^{10}, \ 6.4 \times 10^{15}, \ 2.1 \times 10^{20}, \ 1.7 \times 10^{15}\end{align*}
14. \begin{align*}5.63 \times 10^{-5}, \ 4.16 \times 10^{-3}, \ 3.42 \times 10^{-6}, \ 8.71 \times 10^{-3}\end{align*}
15. \begin{align*}2.12 \times 10^{-5}, \ 0.0000202, \ 2.02 \times 10^{-4}, \ 0.0000221\end{align*}
16. \begin{align*}86,102,000, \ 8.61 \times 10^8, \ 86,120,000, \ 8.61 \times 10^7\end{align*}
Directions: Add, subtract, multiply or divide
17. \begin{align*}266 \times 10^{-4} + 8.6 \times 10^{-6}\end{align*}
18. \begin{align*}7.14 \times 10^4 - 5.5 \times 10^3\end{align*}
19. \begin{align*}(2.56 \times 10^{-3}) \times (3.8 \times 10^6)\end{align*}
20. \begin{align*}(4.97 \times 10^8) \div (7.9 \times 10^5)\end{align*}
Directions: Solve each problem.
21. Mercury’s orbital velocity rounds to about \begin{align*}1.07 \times 10^5\end{align*} miles per hour. Pluto’s orbital velocity rounds to about \begin{align*}1.07 \times 10^4\end{align*} miles per hour. About how many times faster is Mercury than Pluto?
22. The human body uses red blood cells to deliver oxygen to the organs. An average adult has about \begin{align*}2 \times 10^{13}\end{align*} red blood cells. Each cell is \begin{align*}2 \times 10^{-5}\end{align*} meters long. If all the body’s red blood cells were lined up, how many meters would it be? Put your answer in scientific notation and standard form.
23. Jupiter has a mass of \begin{align*}1.898 \times 10^{27} \ kg\end{align*}. Earth has a mass of \begin{align*}5.98 \times 10^{24} \ kg\end{align*}. How much greater is Jupiter’s mass than Earth’s? Put your answer in scientific notation and standard form.