3.8: Scientific Notation

Standard Form to Scientific Form

Very large numbers such as 43,000,000,000,000,000,000 (the number of different possible configurations of Rubik’s cube) and very small numbers such as 0.000000000000000000000340 (the mass of the amino acid tryptophan) are extremely inconvenient to write and read. Such numbers can be expressed more conveniently by writing them as part of a power of 10.

To see how this is done, let us start with a somewhat smaller number such as 2480. Notice that

2480⏟Standard form =248.0×101=24.80×102=2.480×103⏟__⏟__⏟Scientific form 

Scientific Form

The last form is called the scientific form of the number. There is one nonzero digit to the left of the decimal point and the absolute value of the exponent on 10 records the number of places the original decimal point was moved to the left.

0.00059=0.005910=0.0059101=0.0059×10−1=0.059100=0.059102=0.059×10−2=0.591000=0.59103=0.59×10−3=5.910,000=5.9104=5.9×10−4

There is one nonzero digit to the left of the decimal point and the absolute value of the exponent of 10 records the number of places the original decimal point was moved to the right.

Scientific Notation

Numbers written in scientific form are also said to be written using scientific notation. In scientific notation, a number is written as the product of a number between and including 1 and 10 (1 is included,10 is not) and some power of 10.

Writing a Number in Scientific Notation

To write a number in scientific notation:

1. Move the decimal point so that there is one nonzero digit to its left.
2. Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

Sample Set A

Write the numbers in scientific notation.

Example 3.8.1

981

The number 981 is actually 981., and it is followed by a decimal point. In integers, the decimal point at the end is usually omitted.

981 =981. =9.81 ×102

The decimal point is now two places to the left of its original position, and the power of 10 is 2.

Example 3.8.2

54.066 =5.4066 ×101 =5.4066 ×10

The decimal point is one place to the left of its original position, and the power of 10 is 1

Example 3.8.3

0.000000000004632 =4.632 ×10−12

The decimal point is twelve places to the right of its original position, and the power of 10 is −12.

Example 3.8.4

0.027 =2.7 ×10−2

The decimal point is two places to the right of its original position, and the power of 10 is −2

Practice Set A

Write the following numbers in scientific notation.

Practice Problem 3.8.1

346

Answer

3.46 ×102

Practice Problem 3.8.2

72.33

Answer

7.233 ×10

Practice Problem 3.8.3

5387.7965

Answer

5.3877965 ×103

Practice Problem 3.8.4

87,000,000

Answer

8.7 ×107

Practice Problem 3.8.5

179,000,000,000,000,000,000

Answer

1.79 ×1020

Practice Problem 3.8.6

100,000

Answer

1.0 ×105

Practice Problem 3.8.7

1,000,000

Answer

1.0 ×106

Practice Problem 3.8.8

0.0086

Answer

8.6 ×10−3

Practice Problem 3.8.9

0.000098001

Answer

9.8001 ×10−5

Practice Problem 3.8.10

0.000000000000000054

Answer

5.4 ×10−17

Practice Problem 3.8.11

0.0000001

Answer

1.0 ×10−7

Practice Problem 3.8.12

0.00000001

Answer

1.0 ×10−8

Scientific Form to Standard Form

A number written in scientific notation can be converted to standard form by reversing the process shown in Sample Set A.

Converting from Scientific Notation

To convert a number written in scientific notation to a number in standard form, move the decimal point the number of places prescribed by the exponent on the 10.

Positive Exponent Negative Exponent

Move the decimal point to the right when you have a positive exponent, and move the decimal point to the left when you have a negative exponent.

Sample Set B

Example 3.8.5

4.63 ×104

The exponent of 10 is 4 so we must move the decimal point to the right 4 places (adding 0's if necessary).

4.6730 ×104 =46730

Example 3.8.5

2.9 ×107

The exponent of 10 is 7 so we must move the decimal point to the right 7 places (adding 0's if necessary).

2.9 ×107 =29000000

Example 3.8.6

1 ×1027

The exponent of 10 is 27 so we must move the decimal point to the right 27 places (adding 0's if necessary).

1 ×1027 =1,000,000,000,000,000,000,000,000,000

Example 3.8.7

4.21 ×10−5

The exponent of 10 is −5 so we must move the decimal point to the left 5 places (adding 0's if necessary).

4.21 ×10−5 =0.0000421

Example 3.8.8

1.006 ×10−18

The exponent of 10 is −18 so we must move the decimal point to the left 18 places (adding 0's if necessary).

1.006 ×10−18 =0.000000000000000001006

Practice Set B

Convert the following numbers to standard form.

Practice Problem 3.8.13

9.25 ×102

Answer

925

Practice Problem 3.8.14

4.01 ×105

Answer

401000

Practice Problem 3.8.15

1.2 ×10−1

Answer

0.12

Practice Problem 3.8.16

8.88 ×10−5

Answer

0.0000888

Working with Numbers in Scientific Notation

Multiplying Numbers Using Scientific Notation

There are many occasions (particularly in the sciences) when it is necessary to find the product of two numbers written in scientific notation. This is accomplished by using two of the basic rules of algebra.

Suppose we wish to find \((a \times 10^n)(b \times 10^m). Since the only operation is multiplication, we can use the commutative property of multiplication to rearrange the numbers.

\((a \times 10^n)(b \times 10^m) = (a \times b)(10^n \times 10^m)

Then, by the rules of exponents, 10𝑛 ×10𝑚 =10𝑛+𝑚. Thus,

(𝑎 ×10𝑛)⁢(𝑏 ×10𝑚) =(𝑎 ×𝑏) ×10𝑛+𝑚

The product of (𝑎 ×𝑏) may not be between 1 and 10, so (𝑎 ×𝑏) ×10𝑛+𝑚 may not be in scientific form. The decimal point in (𝑎 ×𝑏) may have to be moved. An example of this situation is in Sample Set C, example 3.8.10.

Sample Set C

Example 3.8.9

(2×103)⁢(4×108)=(2×4)⁢(103×108)=8×103+8=8×1011

Example 3.8.10

(5×1017)⁢(8.1×10−22)=(5×8.1)⁢(1017×10−22)=40.5×1017−22=40.5×10−5

We need to move the decimal point one place to the left to put this number in scientific notation.
Thus, we must also change the exponent of 10.

40.5×10−54.05×101×10−54.05×(101×10−5)4.05×(101−5)4.05×10−4

Thus,
(5×1017)⁢(8.1×10−22) =4.05 ×10−4

Practice Set C

Perform each multiplication.

Practice Problem 3.8.17

(3 ×105)⁢(2 ×1012)

Answer

6 ×1017

Practice Problem 3.8.18

(1 ×10−4)(6 ×1024

Answer

6 ×1020

Practice Problem 3.8.19

(5 ×1018)⁢(3 ×106)

Answer

1.5 ×1025

Practice Problem 3.8.20

(2.1 ×10−9)⁢(3 ×10−11)

Answer

6.3 ×10−20

Exercises

Convert the numbers used in the following problems to scientific notation.

Exercise 3.8.1

Mount Kilimanjaro is the highest mountain in Africa. It is 5890 meters high.

Answer

5.89 ×103

Exercise 3.8.2

The planet Mars is about 222,900,000,000 meters from the sun.

Exercise 3.8.3

There is an irregularly shaped galaxy, named NGC 4449, that is about 250,000,000,000,000,000,000,000 meters from earth.

Answer

2.5 ×1023

Exercise 3.8.4

The farthest object astronomers have been able to see (as of 1981) is a quasar named 3C427. There seems to be a haze beyond this quasar that appears to mark the visual boundary of the universe. Quasar 3C427 is at a distance of 110,000,000,000,000,000,000,000,000 meters from the earth.

Exercise 3.8.5

The smallest known insects are about the size of a typical grain of sand. They are about 0.0002 meters in length (2 ten-thousandths of a meter).

Answer

2 ×10−4

Exercise 3.8.6

Atoms such as hydrogen, carbon, nitrogen, and oxygen are about 0.0000000001 meter across.

Exercise 3.8.7

The island of Manhattan, in New York, is about 57,000 square meters in area.

Answer

5.7 ×104

Exercise 3.8.8

The second largest moon of Saturn is Rhea. Rhea has a surface area of about 735,000 square meters, roughly the same surface area as Australia.

Exercise 3.8.9

A star, named Epsilon Aurigae B, has a diameter (distance across) of 2,800,000,000,000 meters. This diameter produces a surface area of about 24,630,000,000,000,000,000,000,000 square meters. This star is what astronomers call a red giant and it is the largest red giant known. If Epsilon Aurigae were placed at the sun’s position, its surface would extend out to the planet Uranus.

Answer

2.8 ×1012, 2.463 ×1025

Exercise 3.8.10

The volume of the planet Venus is 927,590,000,000,000,000,000 cubic meters.

Exercise 3.8.11

The average mass of a newborn American female is about 3360 grams.

Answer

3.36 ×103

Exercise 3.8.12

The largest brain ever measured was that of a sperm whale. It had a mass of 9200 grams.

Exercise 3.8.13

The mass of the Eiffel tower in Paris, France, is 8,000,000 grams.

Answer

8 ×106

Exercise 3.8.14

In 1981, a Japanese company built the largest oil tanker to date. The ship has a mass of about 510,000,000,000 grams. This oil tanker is more than 6 times as massive as the U.S. aircraft carrier, U.S.S. Nimitz.

Exercise 3.8.15

In the constellation of Virgo, there is a cluster of about 2500 galaxies. The combined mass of these galaxies is 150,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 grams.

Answer

1.5 ×1062

Exercise 3.8.16

The mass of an amoeba is about 0.000004 gram.

Exercise 3.8.17

Cells in the human liver have masses of about 0.000000008 gram.

Answer

8 ×10−9

Exercise 3.8.18

The human sperm cell has a mass of about 0.000000000017 gram.

Exercise 3.8.19

The principal protein of muscle is myosin. Myosin has a mass of 0.00000000000000000103 gram.

Answer

1.03 ×10−18

Exercise 3.8.20

Amino acids are molecules that combine to make up protein molecules. The amino acid tryptophan has a mass of 0.000000000000000000000340 gram.

Exercise 3.8.21

An atom of the chemical element bromine has 35 electrons. The mass of a bromine atom is 0.000000000000000000000000031 gram.

Answer

3.1 ×10−26

Exercise 3.8.22

Physicists are performing experiments that they hope will determine the mass of a small particle called a neutrino. It is suspected that neutrinos have masses of about 0.0000000000000000000000000000001 gram.

Exercise 3.8.23

The approximate time it takes for a human being to die of asphyxiation is 316 seconds.

Answer

3.16 ×102

Exercise 3.8.24

On the average, the male housefly lives 1,468,800 seconds (17 days).

Exercise 3.8.25

Aluminum-26 has a half-life of 740,000 years.

Answer

7.4 ×105

Exercise 3.8.26

Manganese-53 has a half-life of 59,918,000,000,000 seconds (1,900,000 years).

Exercise 3.8.27

In its orbit around the sun, the earth moves a distance one and one half feet in about 0.0000316 second.

Answer

3.16 ×10−5

Exercise 3.8.28

A pi-meson is a subatomic particle that has a half-life of about 0.0000000261 second.

Exercise 3.8.29

A subatomic particle called a neutral pion has a half-life of about 0.0000000000000001 second.

Answer

1 ×10−16

Exercise 3.8.30

Near the surface of the earth, the speed of sound is 1195 feet per second.

For the following problems, convert the numbers from scientific notation to standard decimal form.

Exercise 3.8.31

The sun is about 1 ×108 meteres from earth.

Answer

100,000,000

Exercise 3.8.32

The mass of the earth is about 5.98 ×1027 grams.

Exercise 3.8.33

Light travels about 5.866 ×1012 miles in one year.

Answer

5,866,000,000,000

Exercise 3.8.34

One year is about 3 ×107 seconds.

Exercise 3.8.35

Rubik’s cube has about 4.3 ×1019 different configurations.

Answer

43,000,000,000,000,000,000

Exercise 3.8.36

A photon is a particle of light. A 100-watt light bulb emits 1 ×1020 photons every second.

Exercise 3.8.37

There are about 6 ×107 cells in the retina of the human eye.

Answer

60,000,000

Exercise 3.8.38

A car traveling at an average speed will travel a distance about equal to the length of the smallest fingernail in 3.16 ×10−4 seconds.

Exercise 3.8.39

A ribosome of E. coli has a mass of about 4.7 ×10−19 grams.

Answer

0.00000000000000000047

Exercise 3.8.40

A mitochondrion is the energy-producing element of a cell. A mitochondrion is about 1.5 ×10−6 meters in diameter.

Exercise 3.8.41

There is a species of frogs in Cuba that attain a length of at most 1.25 ×10−2 meters.

Answer

0.0125

Perform the following operations.

Exercise 3.8.42

(2 ×104)⁢(3 ×105)

Exercise 3.8.43

(4 ×102)⁢(8 ×106)

Answer

3.2 ×109

Exercise 3.8.44

(6 ×1014)⁢(6 ×10−10)

Exercise 3.8.45

(3 ×10−5)⁢(8 ×107)

Answer

2.4 ×103

Exercise 3.8.46

(2 ×10−1)⁢(3 ×10−5)

Exercise 3.8.47

(9 ×10−5)⁢(1 ×10−11)

Answer

9 ×10−16

Exercise 3.8.48

(3.1 ×104)⁢(3.1 ×10−6)

Exercise 3.8.49

4.2 ×10−12)(3.6 ×10−20)

Answer

1.512 ×10−31

Exercise 3.8.50

(1.1×106)2

Exercises for Review

Exercise 3.8.51

What integers can replace 𝑥 so that the statement −6 <𝑥 <−2 is true?

Answer

−5,−4,−3

Exercise 3.8.52

Simplify (5⁢𝑥2⁢𝑦4)⁢(2⁢𝑥⁢𝑦5)

Exercise 3.8.53

Determine the value of −[−(−|−5|)].

Answer

−5

Exercise 3.8.54

Write \(\dfrac{x^3y^{-5}}{z^{-4}\) so that only positive exponents appear.

Exercise 3.8.55

Write (2⁢𝑧+1)3⁢(2⁢𝑧+1)−5 so that only positive exponents appear.

Answer

1(2⁢𝑧+1)2

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