Contemporary Mathematics

# 3.9Scientific Notation

Contemporary Mathematics3.9 Scientific Notation

Figure 3.45 Calculations in the sciences often involve numbers in scientific notation form.

### Learning Objectives

After completing this section, you should be able to:

1. Write numbers in standard or scientific notation.
2. Convert numbers between standard and scientific notation.
3. Add and subtract numbers in scientific notation.
4. Multiply and divide numbers in scientific notation.
5. Use scientific notation in computing real-world applications.

The amount of information available on the Internet is simply incomprehensible. One estimate for the amount of data that will be on the Internet by 2025 is 175 Zettabytes. A single zettabyte is one billion trillion. Written out, it is 1,000,000,000,000,000,000,000. One estimate is that we’re producing 2.5 quintillion bytes of data per day. A quintillion is a trillion trillion, or, written out, 1,000,000,000,000,000,000. To determine how many days it takes to increase the amount of information that is on the Internet by 1 zettabyte, divide these two numbers, a zettabyte being 1,000,000,000,000,000,000,000, and 2.5 quintillion, being 2,500,000,000,000,000,000, shows it takes 400 days to generate 1 zettabyte of information. But doing that calculation is awkward with a calculator. Keeping track of the zeros can be tedious, and a mistake can easily be made.

On the other end of the scale, a human red blood cell has a diameter of 7.8 micrometers. One micrometer is one millionth of a meter. Written out, 7.8 micrometers is 0.0000078 meters. Smaller still is the diameter of a virus, which is about 100 nanometers in diameter, where a nanometer is a billionth of a meter. Written out, 100 nanometers is 0.0000001 meters. To compare that to engineered items, a single transistor in a computer chip can be 14 nanometers in size (0.000000014 meters). Smaller yet is the diameter of an atom, at between 0.1 and 0.5 nanometers.

Sometimes we have numbers that are incredibly big, and so have an incredibly large number of digits, or sometimes numbers are incredibly small, where they have a large number of digits after the decimal. But using those representations of the names of the sizes makes comparing and computing with these numbers problematic. That’s where scientific notation comes in.

### Writing Numbers in Standard or Scientific Notation Form

When we say that a number is in scientific notation, we are specifying the form in which that number is written. That form begins with an integer with an absolute value between 1 and 9, then perhaps followed the decimal point and then some more digits. This is then multiplied by 10 raised to some power. When the number only has one non-zero digit, the scientific notation form is the digit multiplied by 10 raised to an exponent. When the number has more than one non-zero digit, the scientific notation form is a single digit, followed by a decimal, which is then followed by the remaining digits, which is then multiplied by 10 to a power.

The following numbers are written in scientific notation:

$1.45×1031.45×103$

$−8.345×10−4−8.345×10−4$

$3×1023×102$

$3.14159×1003.14159×100$

The following numbers are not written in scientific notation:

$1.451.45$ because it isn't multiplied by 10 raised to a power

$−50.053×107−50.053×107$ because the absolute value of −50.053 is not at least 1 and less than 10

$41.7×10941.7×109$ because 41.7 is not at least 1 and less than 10

$0.036×10−30.036×10−3$ because 0.036 is not at least one and less than 10

### Example 3.117

#### Identifying Numbers in Scientific Notation

Which of the following numbers are in scientific notation? If the number is not in scientific notation, explain why it is not.

1. $−9.67×1020−9.67×1020$
2. $145×10−8145×10−8$
3. $1.451.45$

Which of the following numbers are in scientific notation? If the number is not in scientific notation, explain why it is not.
1.
${\text{42}}{\text{.67}} \times {10^{13}}$
2.
$7.113 \times {10^{ - 2}}$
3.
$- 80.91$

Some numbers are so large or so small that it is impractical to write them out fully. Avogadro’s number is important in chemistry. It represents the number of units in 1 mole of any substance. The substance many be electrons, atoms, molecules, or something else. Written out, the number is: 602,214,076,000,000,000,000,000. Another example of a number that is impractical to write out fully is the length of a light wave. The wavelength of the color blue is about 0.000000450 to 0.000000495 meters. Such numbers are awkward to work with, and so scientific notation is often used. We need to discuss how to convert numbers into scientific notation, and also out of scientific notation.

Recall that multiplying a number by 10 adds a 0 to the end of the number or moves the decimal one place to the right, as in $43×10=43043×10=430$ or $3.89×10=38.93.89×10=38.9$. And if you multiply by 100, it adds two zeros to the end of the number or moves the decimal two places to the right, and so on. For example, $38×100,000=3,800,00038×100,000=3,800,000$ and $32.998×10,000=329,98032.998×10,000=329,980$. Multiplying a number by 1 followed by some number of zeros just adds that many zeros to the end of the number or moves the decimal place that many places to the right. Numbers written as 1 followed by some zeros are just powers of 10, as in $101=10101=10$, $102=100102=100$, $103=1,000103=1,000$, etc. Generally, $10n=10...0︸n10n=10...0︸n$.

We can use this to write very large numbers. For instance, Avogadro’s number is 602,214,076,000,000,000,000,000, which can be written as $6.02214076×10236.02214076×1023$. The multiplication moves the decimal 23 places to the right.

Similarly, when we divide by 10, we move the decimal one place to the left, as in $46.710=4.6746.710=4.67$. If we divide by 100, we move the decimal two places to the left, as in $3.456100=0.034563.456100=0.03456$. In general, when you divide a number by a 1 followed by $nn$ zeros, you move the decimal $nn$ places to the left, as in $8,244.9021,000,000=0.0082449028,244.9021,000,000=0.008244902$. This denominator could be written as $106106$. If we use that in the expression and allow for negative exponents, rewrite the number as $8,244.9021,000,000=8,244.902106=8,244.902×10−68,244.9021,000,000=8,244.902106=8,244.902×10−6$. With this, we can write division by a 1 followed by $nn$ zeros as multiplication by 10 raised to $‒n‒n$.

Using that information, we can demonstrate how to convert from a number in standard form into scientific notation form.

Case 1: The number is a single-digit integer.

In this case, the scientific notation form of the number is $digit×101digit×101$.

Case 2: The absolute value of the number is less than 1.

• Step 1: Count the number of zeros between the decimal and the first non-zero digit. Label this $nn$.
• Step 2: Starting with the first non-zero digit of the number, write the digits. If the number was negative, include the negative sign.
• Step 3: If there is more than one digit, place the decimal after the first digit from Step 2.
• Step 4: Multiply the number from Step 3 by $10n+110n+1$.

Case 3: The absolute value of the number is 10 or larger.

• Step 1: Count the number of digits that are to the left of the decimal point. Label this $nn$.
• Step 2: Write the digits of the number without the decimal place, if one was present. If the number was negative, include the negative sign.
• Step 3: If there is more than one digit, place the decimal point after the first digit.
• Step 4: Multiply the number from Step 3 by $10n−110n−1$.

### Example 3.118

#### Writing a Number in Scientific Notation

Write the following numbers in scientific notation form:

1. 428.9
2. −0.00000981
3. 8

Write the following numbers in scientific notation form:
1.
−38300
2.
0.0045
3.
1

When we write numbers in scientific notation form, we can manipulate the representation of the number by moving the decimal around, and making an appropriate change to the exponent of the 10. For instance, let’s look at $145.8141×108145.8141×108$. If we wanted to move the decimal one place to the left, we’d have to increase the power of 10, as shown here: $145.8141×108=14.58141×109145.8141×108=14.58141×109$. Since we moved the decimal one to the left, we balance that with moving the exponent up by one. Similarly, if we move the decimal one place to the right, we have to balance that by moving the exponent one to the left, or subtracting one from the exponent, as shown here: $145.8141×108=1458.141×107145.8141×108=1458.141×107$. Generally, for a number in the form $number×10nnumber×10n$:

• If you move the decimal to the left by $kk$ digits, you increase the exponent by $kk$.
• If you move the decimal to the right by $kk$ digits, you decrease the exponent by $kk$ digits.

### Example 3.119

#### Increasing the Exponent

Change $456.142×105456.142×105$ by moving the decimal two places to the left.

1.
Change $46.113 \times {10^8}$ by moving the decimal four places to the left.

### Example 3.120

#### Decreasing the Exponent

Change $12.3×10212.3×102$ by moving the decimal five places to the right.

1.
Change $149.11 \times {10^{ - 4}}$ by moving the decimal two places to the right.

### Converting Numbers from Scientific Notation to Standard Form

In the previous section, converting a number from standard form to scientific notation was explored. Now, we explore converting from scientific notation back into standard form. Doing so involves moving the decimal according to the power of the 10. The decimal is moved a number of steps equal to the exponent of the 10. As demonstrated previously, when the exponent of the 10 is negative, the decimal is moved to the left and when the exponent of the 10 is positive, the decimal is moved to the right.

### Example 3.121

#### Converting from Scientific Notation to Standard Form

Convert the following into standard form:

1. $2.78×1092.78×109$
2. $9.04×10−89.04×10−8$

Convert the following into standard form:
1.
$1.02 \times {10^6}$
2.
$4.09 \times {10^{ - 5}}$

### Tech Check

#### Scientific Notation on a Calculator

Most scientific and graphing calculators come with the ability to directly convert from standard form to scientific notation. On the TI-83, it is accessed through the MODE menus. For a commonly used, free phone scientific calculator, the calculator can be forced to work in scientific notation mode through its settings.

Some calculators, such as the Desmos online calculator, display scientific notation as a number times 10 to a power as you’ve seen in this section. However, some calculators indicate scientific notation by replacing the $×10n×10n$ with an E (or EE) followed by the exponent. For example, Figure 3.46 shows what you may see on a TI-84.

Figure 3.46 Calculator screens

### Adding and Subtracting Numbers in Scientific Notation

To add or subtract numbers in scientific notation, the numbers first need to have the same exponent for the 10s. It is possible to add the following since the powers of 10 match: $4.5×104+3.15×104=7.65×1044.5×104+3.15×104=7.65×104$

Notice that the number parts were added, but the exponent part remained the same. This is due to the distributive property of the real numbers. The $104104$ is factored from the two terms, as shown: $4.5×104+3.15×104=(4.5+3.15)×104=7.65×1044.5×104+3.15×104=(4.5+3.15)×104=7.65×104$

Numbers in scientific notation can be added or subtracted directly using a calculator. Simply enter the values in scientific form and set your calculator to display scientific notation.

### Example 3.122

#### Adding and Subtracting Numbers in Scientific Notation with the Same Powers of 10

Calculate the following:

1. $3.8×10−3+1.006×10−33.8×10−3+1.006×10−3$
2. $9.61×108−3.85×1089.61×108−3.85×108$

Calculate the following:
1.
$7.57 \times {10^{13}} + 2.031 \times {10^{13}}$
2.
$3.03 \times {10^{ - 6}} - 1.5 \times {10^{ - 6}}$

Adding and subtracting in scientific notation is straightforward when the exponents are the same. There are two issues that can arise. The first issue is what to do if after adding or subtracting the result is not in scientific notation.

### Example 3.123

Calculate the following:

1. $7.03×1013+8.5×10137.03×1013+8.5×1013$
2. $4.3×1021−4.613×10214.3×1021−4.613×1021$

Calculate the following:
1.
$5.08 \times {10^3} + 6.9 \times {10^3}$
2.
$8.968 \times {10^{ - 38}} - 8.761 \times {10^{ - 38}}$

The second issue that might be encountered when adding or subtracting is that the powers of 10 do not match. In that case, one of the numbers must be changed so that the powers of 10 match. It is easiest to make the smaller power of 10 larger to match the other power of 10.

For example, to perform the following, $4.5×105+3.9×1034.5×105+3.9×103$, we’d change the $3.9×1033.9×103$ so that the power of 10 is 5. To do so, we need to increase the power of 10 and move the decimal in the number part two places to the left. That would alter $3.9×1033.9×103$ into $0.039×1050.039×105$. We would use $0.039×1050.039×105$ in the addition problem, so that the exponents match, allowing the addition to occur. $4.5×105+3.9×103=4.5×105+0.039×105=(4.5+0.039)×105=4.539×1054.5×105+3.9×103=4.5×105+0.039×105=(4.5+0.039)×105=4.539×105$

The steps to take when the exponents of the 10s are not equal are:

Step 1: Increase the smaller exponent to equal the larger exponent. Label the amount increased as $nn$.

Step 2: For the number with the smaller power of 10, move the decimal point of the number part to the left $nn$ places.

Step 3: Perform the addition or subtraction.

Step 4: If the result is not in scientific notation, adjust the number to be in scientific notation.

### Example 3.124

#### Adding Numbers in Scientific Notation with Different Powers of 10

Calculate the following:

$6.1×104+4.8×1056.1×104+4.8×105$

1.
Calculate the following:
$1.14 \times {10^{ - 43}} + 2.56\,{ \times\, 10^{ - 46}}$

### Example 3.125

#### Subtracting Numbers in Scientific Notation with Different Powers of 10

Calculate the following:
$7.9×10−15−6.8×10−137.9×10−15−6.8×10−13$

1.
Calculate the following:
$9.15 \times {10^{28}} - 7.23 \times {10^{26}}$

### Multiplying and Dividing Numbers in Scientific Notation

Multiplying and dividing numbers in scientific notation is somewhat easier than adding or subtracting, because the exponents of the 10s do not have to match. However, it is much more likely that the result will not be in scientific notation, and so that will have to be adjusted at the end. Generally, we multiply or divide the number parts of the two values, and then apply exponent rules to the 10 raised to the powers.

To multiply two numbers in scientific notation:

Step 1: Multiply the number parts.

Step 2: Add the exponents of the 10s.

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

Step 4: If the number is not in scientific notation, adjust it appropriately.

### Example 3.126

#### Multiplying Numbers in Scientific Notation

Calculate the following:

1. $(4.3×103)×(1.8×107)(4.3×103)×(1.8×107)$

2. $(5×10−13)×(7.3×106)(5×10−13)×(7.3×106)$

Calculate the following:
1.
$(2.29 \times {10^3}) \times (3 \times {10^4})$
2.
$(6.91 \times {10^{ - 3}}) \times (9.1 \times {10^5})$

### Dividing Numbers in Scientific Notation

To divide two numbers that are in scientific notation:

Step 1: Divide the number parts.

Step 2: Subtract the exponent of the denominator from the exponent of the numerator.

Step 3: The answer is the result from Step 1 times 10 raised to the result from Step 2.

Step 4: If the number is not in scientific notation, adjust it appropriately.

### Example 3.127

#### Dividing Numbers in Scientific Notation

Calculate the following:

1. $(8.4×1031)/(2.1×107)(8.4×1031)/(2.1×107)$
2. $(4.14×10−13)/(8.28×109)(4.14×10−13)/(8.28×109)$

Calculate the following:
1.
$(3.6 \times {10^{ - 2}})/(1.5 \times {10^3})$
2.
$(1.8 \times {10^4})/(4.8 \times {10^3})$

### Using Scientific Notation in Computing Real-World Applications

As noted at the start of this section, scientific notation is useful when the standard representation of a number is awkward or impractical, which occurs when the numbers being used are extremely large or extremely small. For example, Venus is 67,667,000 miles from the sun. In scientific notation, this is $6.7667×1076.7667×107$. Planetary and galaxy distances is one set of numbers that is easier to express using scientific notation.

### Example 3.128

#### Calculating Distances

How much farther from the sun is Earth compared to Venus if Venus is $6.7667×1076.7667×107$ miles from the sun and Earth is $9.1692×1079.1692×107$ miles from the sun?

1.
Earlier we saw that a single transistor in a computer chip 0.000000014 meters, or $1.4 \times {10^{ - 8}}$ m, in size, and that the diameter of an atom could be 0.2 nanometers, or $2 \times {10^{ - 10}}$ m in size. How much larger is the transistor than the atom?

### Example 3.129

#### Calculating Probability

The probability of winning the Mega Millions lottery is published as $3.304693×10−93.304693×10−9$. The probability of being hit by lightning is approximated to be $2×10−62×10−6$. How many times more likely are you to be hit by lightning than win the Mega Millions?

1.
Mercury is about $3.114 \times {10^7}$ miles from the sun. Neptune is about $2.781 \times {10^9}$ miles from the sun. How many times further is Neptune from the sun than Mercury?

### Example 3.130

#### Calculating Time and Length

Sometimes it is entertaining to determine the time it takes for something to happen. Fingernails grow about $8.032×10−118.032×10−11$ km per minute. How many kilometers long would fingernails be after $6×1046×104$ minutes?

1.
There are approximately $1 \times {10^{12}}$ grains of sand in a cubic meter. If the number of grains of sand on the Australian coastline is roughly $7.5 \times {10^{21}}$ grains, roughly how many cubic meters of sand is there on the Australian coastline?

### Example 3.131

#### Calculating Data Generated

As mentioned in the opening to this section, it is estimated that we’re producing 2.5 quintillion bytes of data per day. A good estimate is that there are 7.674 billion people on the planet. Convert both of those numbers to scientific notation, and then determine how much data is being generated per person each day.

1.
Humans collectively exhale approximately $6.4235 \times {10^{12}}$ pounds of carbon dioxide per year. There are approximately $7.647 \times {10^9}$ humans currently living on Earth. How many pounds of carbon dioxide does a single human, on average, exhale per year?

### What Numbers Could Be Considered “Too Big” or “Too Small”?

One wonders when the numbers we represent become too large or small for consideration. Perhaps the following examples put limits on what is meaningful. The number of particles in the known universe has been estimated at $4×10804×1080$ particles. The smallest distance that has been measured is $1×10−18m1×10−18m$, though the theoretical smallest measurable value is $1×10−35m1×10−35m$. The distance across the universe is $4.4×1026m4.4×1026m$. Considering what those numbers represent, the extreme largest and extreme smallest, they might be numbers that constrain what we should reasonably be expected to deal with.

45.
Write 0.00456 in scientific notation.
46.
Write $5.67 \times {10^8}$ in standard form.
47.
Calculate $4.5 \times {10^3} + 9.8 \times {10^2}$.
48.
Calculate $2.5 \times {10^5} - 9.8 \times {10^6}$.
49.
Calculate $(7.4 \times {10^4}) \times (4.8 \times {10^3})$.
50.
Calculate $\frac{{4.6 \times {{10}^{ - 4}}}}{{8 \times {{10}^{ - 8}}}}$.
51.
The distance from Earth to the moon is $1.514 \times {10^{10}}$ inches. The thickness of a dollar bill is $4.3 \times {10^{ - 3}}$ inches. How many dollar bills must be stacked so the pile reaches the moon?

### Section 3.9 Exercises

For the following exercises, convert numbers to scientific notation.
1 .
$0.0134$
2 .
$0.0000761$
3 .
$3,400$
4 .
$8,980,000$
For the following exercises, convert numbers to standard form.
5 .
$9.01 \times {10^5}$
6 .
$3.78 \times {10^7}$
7 .
$4.32 \times {10^{ - 3}}$
8 .
$5.781 \times {10^{ - 5}}$
For the following exercises, the numbers are not in scientific notation. Convert them to scientific notation.
9 .
$37.65 \times {10^4}$
10 .
$0.0034 \times {10^6}$
11 .
$0.0834 \times {10^{ - 7}}$
12 .
$14.56 \times {10^{ - 3}}$
For the following exercises, make the conversions required.
13 .
The distance from the sun to the star Polaris is about 3,056,000,000,000,000 km. Express that distance in scientific notation.
14 .
The distance from us to the next-closest galaxy is about 662,000,000,000,000,000 km. Express that distance in scientific notation.
15 .
The mass of a grain of sand is about $6.66 \times {10^{ - 4}}$ g. Convert that mass to standard form.
16 .
The diameter of a cell is about $2 \times {10^{ - 6}}$ m. Convert that diameter to standard form.
17 .
The equatorial circumference of Earth is approximately $4.007 \times {10^4}$ km. Convert that circumference to standard form.
18 .
The straight-line distance from Buffalo, NY, to Buenos Aires, Argentina, is approximately $8.86 \times {10^6}$ m. Convert that distance to standard form.
19 .
The mass of a proton is approximately $1.67 \times {10^{ - 27}}$ kg. Convert that mass to standard form.
20 .
The diameter of a housefly egg is approximately $1.2 \times {10^{ - 3}}$ m. Convert that diameter to standard form.
21 .
The tallest building in the world, the Burj Khalifa in Dubai, stands at 829.8 m tall. Convert that height to scientific notation.
22 .
Using the rings of the shell, the age of an Icelandic clam is 507 years. Express that age in scientific notation.
Calculate the following:
23 .
$1.3 \times {10^2} + 3.8 \times {10^2}$
24 .
$7.8 \times {10^{12}} + 1.1 \times {10^{12}}$
25 .
$3.36 \times {10^4} + 2.71 \times {10^4}$
26 .
$4.58 \times {10^9} + 1.93 \times {10^9}$
27 .
$8.1 \times {10^{ - 17}} + 1.6 \times {10^{ - 17}}$
28 .
$4.506 \times {10^{ - 3}} + 3.908 \times {10^{ - 3}}$
29 .
$8.602 \times {10^{ - 25}} + 1.096 \times {10^{ - 25}}$
30 .
$2.0557 \times {10^{ - 6}} + 1.001 \times {10^{ - 6}}$
31 .
$5.2 \times {10^4} - 4.1 \times {10^4}$
32 .
$9.48 \times {10^{15}} - 6.78 \times {10^{15}}$
33 .
$7.81 \times {10^{ - 7}} - 4.62 \times {10^{ - 7}}$
34 .
$4.53 \times {10^{ - 3}} - 2.79 \times {10^{ - 3}}$
35 .
$4.6 \times {10^{ - 5}} + 9.1 \times {10^{ - 5}}$
36 .
$6.7 \times {10^8} + 5.7 \times {10^8}$
37 .
$4.13 \times {10^{ - 4}} + 7.93 \times {10^{ - 4}}$
38 .
$5.671 \times {10^{ - 8}} + 9.073 \times {10^{ - 8}}$
39 .
$4.513 \times {10^6} + 7.856 \times {10^4}$
40 .
$7.135 \times {10^8} + 5.143 \times {10^6}$
41 .
$3.17 \times {10^{ - 3}} + 5.92 \times {10^{ - 5}}$
42 .
$4.503 \times {10^{ - 6}} + 3.119 \times {10^{ - 4}}$
43 .
$(4.5 \times {10^4}) \times (1.2 \times {10^6})$
44 .
$(3.45 \times {10^7}) \times (2.81 \times {10^{ - 3}})$
45 .
$(3.1 \times {10^8}) \times (2.7 \times {10^{ - 5}})$
46 .
$(6.32 \times {10^{ - 4}}) \times (1.31 \times {10^{ - 5}})$
47 .
$(3.91 \times {10^6}) \times (8.13 \times {10^2})$
48 .
$(7.12 \times {10^{ - 11}}) \times (6.61 \times {10^{ - 5}})$
49 .
$(3.45 \times {10^4}) \div (1.5 \times {10^6})$
50 .
$(1.4 \times {10^3}) \div (5.6 \times {10^{ - 5}})$
For the following exercises, apply your understanding of scientific notation to real-world applications.
51 .
When stretched out, a strand of human DNA is, on average, $2.066 \times {10^2}$ cm. One centimeter, or 1 cm, is $1 \times {10^{ - 5}}$ km. Determine the average length of a strand of human DNA in kilometers.
52 .
One approximation of the average number of cells in the human body is $3 \times {10^{13}}$ cells (30 trillion!!!). If the DNA of each cell were stretched out and laid end to end, what would be the total length of the DNA in km? Use your answer from Exercise 51 for the length, in kilometers, of DNA.
53 .
The equatorial circumference of Earth is approximately $4.007 \times {10^4}$ km. Use the answer from Exercise 52 to determine the number of times that the stretched out human DNA would encircle Earth.
54 .
The average stride length of a 1.651 m tall woman is $6.6 \times {10^{ - 1}}$ meters. If such a person could walk from Buffalo, NY, to Buenos Aires, Argentina, in a straight line, how many steps would that person need to take? See Exercise 18 for the distance from Buffalo, NY, to Buenos Aires, Argentina.
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