{{ tocSubheader }}
{{ 'ml-label-loading-course' | message }}
An error ocurred, try again later!
Chapter {{ article.chapter.number }}
{{ article.number }}.
{{ article.displayTitle }}
{{ article.intro.summary }}
Show less Show more expand_more Lesson Settings & Tools
| | {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| | {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| | {{ 'ml-lesson-time-estimation' | message }} |
Scientists and astronomers make calculations using huge and tiny numbers. They could go all the way up to 100000000000 or as low as 0.00000000001. Writing these numbers can take a lot of time while making calculations. A more practical way of writing numbers exists called scientific notation. This concept will be explored in this lesson.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Distance From the Earth to the Sun
The distance from the Earth to the Sun is about 150000000 kilometers. 
An astronomer wants to find out how long a spaceship would take to fly to the Sun. He will need to write the given distance several times to make this calculation. Is there a way to make the given number shorter? If yes, how?
Discussion
Scientific Notation
Scientific notation is a compact way of writing very large or very small numbers. A number written in scientific notation is expressed as a product of two numbers.
Similarly, for numbers less than 1, such as 0.000022, the decimal will move from left to right to make the number greater than or equal to 1 and less than 10. In this case, the number of places moved indicates the negative exponent to be used for the base 10 power.
Scientific notation is not only a convenient way to express cumbersome numbers. It also eases the comparison of numerical order of magnitude. For example, it may be difficult to determine how much larger 237000000 is compared to 4530000. However, it is easier to see that 2.37×108 and 4.5×106 differ by a factor of about 102=100.
a×10b
In this form, the first factor is greater than or equal to 1 and less than 10. In other words, it needs to be in the interval [1,10). The second factor is a power of 10 where b is an integer. For example, the number 4 million can be rewritten as the product of 4 and a multiple of 10. Then, the multiple of 10 is rewritten as a base 10 power.
4000000=4×1000000=4×106
Very small decimal numbers can also be written in scientific notation. Consider a number where there are many zeros before the significant figures. Take as 0.000342 as an example.
0.000342=3.42÷10000=3.42×10-4
In such cases, numbers are expressed as a division by a multiple of 10. Division by a multiple of 10 is equivalent to multiplication by a base 10 power with a negative exponent. Consider a few more examples of numbers written in scientific notation. | Decimal Form | Written as a Product or Division Expression | Scientific Notation |
|---|---|---|
| 4505 | 4.505×1000 | 4.505×103 |
| 8320000 | 8.32×1000000 | 8.32×106 |
| 0.0005 | 5÷10000 | 5×10-4 |
| 0.0521 | 5.21÷100 | 5.21×10-2 |
Intuitive Method: Rewriting a Number in Scientific Notation
An intuitive method to rewrite a number into scientific notation is to count the number of places the decimal needs tomove. Consider a number greater than 10. The decimal would move from right to left to make the number less than 10 but still greater than 1. The number of places the decimal moved indicates the positive exponent to be used for the base 10 power.
Example
Rewriting the Heights of Mountains
Ramsha and her friends did a research project about the highest mountains in the world. They found that Mount Everest is the world's highest mountain. It is located in the Himalayan mountain range. They also learned that Mount Elbrus is the highest peak of Europe. It is in southwestern Russia.
a The peak of Mount Everest is at 8848 meters above the sea level. Rewrite this height in scientific notation.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"meters","answer":{"text":["8.848 \\times 10^3"]}}
b Mount Elbrus is an extinct volcano with twin cones that reach 1.851×104 feet. Rewrite this height in standard form.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"feet","answer":{"text":["18510"]}}
Solution
a The height of Mount Everest is 8848 meters. Notice that this number is four digits long when written in standard form. Start by placing the decimal point after the first non-zero digit to write it in scientific notation.
8.848↑First nonzero digit
Next, count the digits after the decimal point.
8.848↑Digits after the decimal point
There are three digits after the decimal point. This number will be written as the exponent of 10.
Standard Form 8848 Scientific Notation8.848×103
b This time the number is given in the scientific notation and the task is to rewrite it into standard form.
1.851×104
The power of 10 is 4. It has a positive exponent. Therefore, the decimal point will be moved to the right 4 times. In other words, the first factor 1.851 will be multiplied by 10 four times. 
1.851=1.8510000…
In this case, there are already three nonzero digits. Only one zero needs to be added to the end of the number. Now, the number is in standard form as a five-digit number.
Scientific Notation 1.851×104 Standard Form18510
Example
Rewriting the Lengths of Bacteria
Ramsha and her friends continue to enjoy their school studies. Next is biology class! They use a microscope to examine two types of bacteria, Escherichia coli and Salmonella.
a The length of an E. coli bacterium is two micrometers long. That is equal to 0.000002 meters. Write this length in scientific notation in meters.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"meters","answer":{"text":["2 \\times 10^{-6}"]}}
b The length of a salmonella bacterium is 1.5×10-6 meters. Rewrite this length in standard form.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"meters","answer":{"text":["0.0000015"]}}
Hint
a Start by placing the decimal point after the first non-zero digit.
Solution
a The length of an Escherichia coli bacterium is given as 0.000002 meters. The goal is to write this number in scientific notation. Begin by placing the decimal point after the first non-zero digit.
0.0 0 0 0 0 2. ↑First nonzero digit
Then, determine the power of 10. That is done by counting the number of digits before the new decimal point. 0.0 0 0 0 0 2.↑Digits before the new decimal point
There are six digits before the decimal point. The given number 0.000002 is less than 1. That means the exponent will be negative.
Standard Form 0.000002 Scientific Notation2×10-6
b The length of a salmonella bacterium is given in scientific notation.
1.5×10-6
Notice that the power of 10 is negative. This means that the decimal point will be moved to the left six times. 
Scientific Notation 1.5×10-6 Standard Form0.0000015
Pop Quiz
Translating Between Scientific Notation and Standard Form
Rewrite the given expression in standard form if it is given in scientific notation. Or rewrite it in scientific notation if it is given in standard form.
Example
Examining the Populations of Some Countries
Ramsha's school has an interactive electronic map that shows population data around the world! Ramsha's geography teacher asked her class to examine the populations of some countries.
They used this map to see the numbers of people living in Brazil, Turkey, China, Australia, and the United States of America.
Now all the rounded numbers can be rewritten as a single digit times a power of 10. Count the zeros to determine the power of 10 for each number. For instance, the rounded population of the USA can be rewritten in this way.
Examine the powers of 10. Begin with identifying the largest power. China has the largest. That means it has the greatest population. Now compare the numbers with the same power of 10. Start with 108 because 108 is greater than 107.
a Write the number of people living in these countries in scientific notation by rounding the given numbers to the greatest place value.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"USA:","formTextAfter":"people","answer":{"text":["3 \\times 10^8"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Brazil:","formTextAfter":"people","answer":{"text":["2 \\times 10^8"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Turkey:","formTextAfter":"people","answer":{"text":["8 \\times 10^7"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"China:","formTextAfter":"people","answer":{"text":["1 \\times 10^9"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Australia:","formTextAfter":"people","answer":{"text":["3 \\times 10^7"]}}
b Sort the countries from greatest to least population.
{"type":"sort","form":{"alts":[{"id":0,"text":"China"},{"id":1,"text":"USA"},{"id":2,"text":"Brazil"},{"id":3,"text":"Turkey"},{"id":4,"text":"Australia"}]},"formTextBefore":"","formTextAfter":"","answer":[0,1,2,3,4]}
Solution
a The populations of the five countries will be written in scientific notation one at a time. Start by rounding all the numbers to the greatest place value in a table.
| Population | Rounded | |
|---|---|---|
| USA | 331002651 | 300000000 |
| Brazil | 212559417 | 200000000 |
| Turkey | 84339067 | 80000000 |
| China | 1439323776 | 1000000000 |
| Australia | 25499884 | 30000000 |
300000000=3×108
Apply the same method so the rounded populations of other countries in standard form can be rewritten in scientific notation. | Population | Rounded | Scientific Notation | |
|---|---|---|---|
| USA | 331002651 | 300000000 | 3×108 |
| Brazil | 212559417 | 200000000 | 2×108 |
| Turkey | 84339067 | 80000000 | 8×107 |
| China | 1439323776 | 1000000000 | 1×109 |
| Australia | 25499884 | 30000000 | 3×107 |
b It is time to determine which country has the greatest population! This process requires comparing the numbers written in scientific notation from Part A.
| Country | Population |
|---|---|
| USA | 3×108 |
| Brazil | 2×108 |
| Turkey | 8×107 |
| China | 1×109 |
| Australia | 3×107 |
USA3×108Brazil2×108
The powers of 10 for the US and Brazil are equal. That means their first factors should be compared. The value 3 is larger than 2. That means the population of the USA is greater than Brazil. Next, compare the populations of Turkey and Australia.
Turkey8×107Australia3×107
Again, the powers of 10 are the same. This indicates that the values of 8 and 3 should be checked. Well, 8 is greater than 3. That means the population of Turkey is greater than Australia. Now the countries can be sorted from greatest to least population.
ChinaUSABrazilTurkeyAustralia1×1093×1082×1088×1073×107
Discussion
Multiplying Numbers in Scientific Notation
Numbers written in scientific notation can be multiplied by using the properties of exponents. Two numbers written in scientific notation a×10b and c×10d can be multiplied by using the Commutative Property of Multiplication and the Product of Powers Property.
(a×10b)×(c×10d)=ac×10b+d
(1.5×102)×(12×105)
Three steps can be followed to multiply these numbers.
1
Write Each Number in Scientific Notation
expand_more It is helpful to check whether both of the numbers are written in scientific notation. Recall that a number is in scientific notation if it is written as a product of two numbers. One of these numbers should be a power of 10. The other number should be greater than or equal to 1 and less than 10.
First Factor1.512×××Second Factor102105✓×
The first number is in scientific notation. The second number is not. Its first factor 12 is greater than 10. Note that 12 can also be expressed as 12.0. The decimal point needs to move one unit to the left for the first factor to become less than 10. The second factor's power of 10 is then increased by that number of moves, 1.
12×105⇔1.2×105+1⇔1.2×106✓
2
Multiply the First Factors
expand_more Both numbers are now written in scientific notation. The first factors of both numbers can be multiplied by using the Commutative Property of Multiplication.
(1.5×102)×(1.2×106)
RemovePar
Remove parentheses
1.5×102×1.2×106
CommutativePropMult
Commutative Property of Multiplication
1.5×1.2×102×106
Multiply
Multiply
1.8×102×106
3
Multiply the Second Factors
expand_more Next, the powers with base 10 are multiplied. Remember to use the Product of Powers Property — exponents are added when multiplying powers with the same base.
4
Write the Result in Scientific Notation
expand_more The result is 1.8×108. The first factor is greater than 1 and less than 10. The second factor is written as a power of 10. Therefore, the result is already in scientific notation.
Discussion
Dividing Numbers in Scientific Notation
Numbers written in scientific notation can be divided by using the properties of exponents. Use the Quotient of Powers Property to divide numbers written in scientific notation.
c×10da×10b=ca×10b−d
12000.36×1023
These numbers written in scientific notation, and they can be divided in four steps. It is similar to the process of multiplying numbers written in scientific notation.
1
Write Each Number in Scientific Notation
expand_more Start by checking whether each number is written in scientific notation.
First Factor0.36××Second Factor1023×
The first factor is less than 1. That means the dividend still needs to be written in scientific notation. Move the decimal point to the right until 0.36 is greater than 1 and less than 10. Then, decrease the second factor's exponent by the number of moves. 0.36×1023⇔3.6×1022
Next, consider the divisor. The divisor 1200 is a whole number. Note that 1200 can also be expressed as 1200.0. Now, move the decimal point three units to the left to get a first factor that is less than 10 and greater than 1. The second factor can then be written as 10 to the power of 3.