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
\(\begin{aligned}
\underbrace{2480}_{\text {Standard form }} &=248.0 \times 10^{1} \\
&=24.80 \times 10^{2} \\
&=\underbrace{2.480 \times 10^{3}}_{\text {Scientific form }}
\end{aligned}\)
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.
\(\begin{aligned}
0.00059 &=\dfrac{0.0059}{10}=\dfrac{0.0059}{10^{1}}=0.0059 \times 10^{-1} \\
&=\dfrac{0.059}{100}=\dfrac{0.059}{10^{2}}=0.059 \times 10^{-2} \\
&=\dfrac{0.59}{1000}=\dfrac{0.59}{10^{3}}=0.59 \times 10^{-3} \\
&=\dfrac{5.9}{10,000}=\dfrac{5.9}{10^{4}}=5.9 \times 10^{-4}
\end{aligned}\)
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 \(\PageIndex{1}\)
\(981\)
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.
\(981 = 981. = 9.81 \times 10^2\)
The decimal point is now two places to the left of its original position, and the power of \(10\) is \(2\).
Example \(\PageIndex{2}\)
\(54.066 = 5.4066 \times 10^1 = 5.4066 \times 10\)
The decimal point is one place to the left of its original position, and the power of \(10\) is \(1\)
Example \(\PageIndex{3}\)
\(0.000000000004632 = 4.632 \times 10^{-12}\)
The decimal point is twelve places to the right of its original position, and the power of \(10\) is \(−12\).
Example \(\PageIndex{4}\)
\(0.027 = 2.7 \times 10^{-2}\)
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 \(\PageIndex{1}\)
\(346\)
- Answer
-
\(3.46 \times 10^2\)
Practice Problem \(\PageIndex{2}\)
\(72.33\)
- Answer
-
\(7.233 \times 10\)
Practice Problem \(\PageIndex{3}\)
\(5387.7965\)
- Answer
-
\(5.3877965 \times 10^3\)
Practice Problem \(\PageIndex{4}\)
\(87,000,000\)
- Answer
-
\(8.7 \times 10^7\)
Practice Problem \(\PageIndex{5}\)
\(179,000,000,000,000,000,000\)
- Answer
-
\(1.79 \times 10^{20}\)
Practice Problem \(\PageIndex{6}\)
\(100,000\)
- Answer
-
\(1.0 \times 10^5\)
Practice Problem \(\PageIndex{7}\)
\(1,000,000\)
- Answer
-
\(1.0 \times 10^6\)
Practice Problem \(\PageIndex{8}\)
\(0.0086\)
- Answer
-
\(8.6 \times 10^{-3}\)
Practice Problem \(\PageIndex{9}\)
\(0.000098001\)
- Answer
-
\(9.8001 \times 10^{-5}\)
Practice Problem \(\PageIndex{10}\)
\(0.000000000000000054\)
- Answer
-
\(5.4 \times 10^{-17}\)
Practice Problem \(\PageIndex{11}\)
\(0.0000001\)
- Answer
-
\(1.0 \times 10^{-7}\)
Practice Problem \(\PageIndex{12}\)
\(0.00000001\)
- Answer
-
\(1.0 \times 10^{-8}\)
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 \(\PageIndex{5}\)
\(4.63 \times 10^4\)
The exponent of \(10\) is \(4\) so we must move the decimal point to the right \(4\) places (adding 0's if necessary).
\(4.6730 \times 10^4 = 46730\)
Example \(\PageIndex{5}\)
\(2.9 \times 10^7\)
The exponent of \(10\) is \(7\) so we must move the decimal point to the right \(7\) places (adding 0's if necessary).
\(2.9 \times 10^7 = 29000000\)
Example \(\PageIndex{6}\)
\(1 \times 10^{27}\)
The exponent of \(10\) is \(27\) so we must move the decimal point to the right \(27\) places (adding 0's if necessary).
\(1 \times 10^{27}= 1,000,000,000,000,000,000,000,000,000\)
Example \(\PageIndex{7}\)
\(4.21 \times 10^{-5}\)
The exponent of \(10\) is \(-5\) so we must move the decimal point to the left \(5\) places (adding 0's if necessary).
\(4.21 \times 10^{-5} = 0.0000421\)
Example \(\PageIndex{8}\)
\(1.006 \times 10^{-18}\)
The exponent of \(10\) is \(-18\) so we must move the decimal point to the left \(18\) places (adding 0's if necessary).
\(1.006 \times 10^{-18} = 0.000000000000000001006\)
Practice Set B
Convert the following numbers to standard form.
Practice Problem \(\PageIndex{13}\)
\(9.25 \times 10^2\)
- Answer
-
\(925\)
Practice Problem \(\PageIndex{14}\)
\(4.01 \times 10^5\)
- Answer
-
\(401000\)
Practice Problem \(\PageIndex{15}\)
\(1.2 \times 10^{-1}\)
- Answer
-
\(0.12\)
Practice Problem \(\PageIndex{16}\)
\(8.88 \times 10^{-5}\)
- 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, \(10^n \times 10^m = 10^{n+m}\). Thus,
\((a \times 10^n)(b \times 10^m) = (a \times b) \times 10^{n+m}\)
The product of \((a \times b)\) may not be between \(1\) and \(10\), so \((a \times b) \times 10^{n+m}\) may not be in scientific form. The decimal point in \((a \times b)\) may have to be moved. An example of this situation is in Sample Set C, example 3.8.10.
Sample Set C
Example \(\PageIndex{9}\)
\(\begin{aligned}
\left(2 \times 10^{3}\right)\left(4 \times 10^{8}\right) &=(2 \times 4)\left(10^{3} \times 10^{8}\right) \\
&=8 \times 10^{3+8} \\
&=8 \times 10^{11}
\end{aligned}\)
Example \(\PageIndex{10}\)
\(
\begin{aligned}
\left(5 \times 10^{17}\right)\left(8.1 \times 10^{-22}\right) &=(5 \times 8.1)\left(10^{17} \times 10^{-22}\right) \\
&=40.5 \times 10^{17-22} \\
&=40.5 \times 10^{-5}
\end{aligned}
\)
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 \(10\).
\(
\begin{array}{l}
40.5 \times 10^{-5} \\
4.05 \times 10^{1} \times 10^{-5} \\
4.05 \times\left(10^{1} \times 10^{-5}\right) \\
4.05 \times\left(10^{1-5}\right) \\
4.05 \times 10^{-4}
\end{array}
\)
Thus,
\(
\left(5 \times 10^{17}\right)\left(8.1 \times 10^{-22}\right)=4.05 \times 10^{-4}
\)
Practice Set C
Perform each multiplication.
Practice Problem \(\PageIndex{17}\)
\((3 \times 10^5)(2 \times 10^{12})\)
- Answer
-
\(6 \times 10^{17}\)
Practice Problem \(\PageIndex{18}\)
\((1 \times 10^{-4})(6 \times 10^{24}\)
- Answer
-
\(6 \times 10^{20}\)
Practice Problem \(\PageIndex{19}\)
\((5 \times 10^{18})(3 \times 10^6)\)
- Answer
-
\(1.5 \times 10^{25}\)
Practice Problem \(\PageIndex{20}\)
\((2.1 \times 10^{-9})(3 \times 10^{-11})\)
- Answer
-
\(6.3 \times 10^{-20}\)
Exercises
Convert the numbers used in the following problems to scientific notation.
Exercise \(\PageIndex{1}\)
Mount Kilimanjaro is the highest mountain in Africa. It is 5890 meters high.
- Answer
-
\(5.89 \times 10^3\)
Exercise \(\PageIndex{2}\)
The planet Mars is about 222,900,000,000 meters from the sun.
Exercise \(\PageIndex{3}\)
There is an irregularly shaped galaxy, named NGC 4449, that is about 250,000,000,000,000,000,000,000 meters from earth.
- Answer
-
\(2.5 \times 10^{23}\)
Exercise \(\PageIndex{4}\)
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 \(\PageIndex{5}\)
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
-
\(2 \times 10^{-4}\)
Exercise \(\PageIndex{6}\)
Atoms such as hydrogen, carbon, nitrogen, and oxygen are about 0.0000000001 meter across.
Exercise \(\PageIndex{7}\)
The island of Manhattan, in New York, is about 57,000 square meters in area.
- Answer
-
\(5.7 \times 10^4\)
Exercise \(\PageIndex{8}\)
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 \(\PageIndex{9}\)
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
-
\(2.8 \times 10^{12}\), \(2.463 \times 10^{25}\)
Exercise \(\PageIndex{10}\)
The volume of the planet Venus is 927,590,000,000,000,000,000 cubic meters.
Exercise \(\PageIndex{11}\)
The average mass of a newborn American female is about 3360 grams.
- Answer
-
\(3.36 \times 10^3\)
Exercise \(\PageIndex{12}\)
The largest brain ever measured was that of a sperm whale. It had a mass of 9200 grams.
Exercise \(\PageIndex{13}\)
The mass of the Eiffel tower in Paris, France, is 8,000,000 grams.
- Answer
-
\(8 \times 10^6\)
Exercise \(\PageIndex{14}\)
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 \(\PageIndex{15}\)
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
-
\(1.5 \times 10^{62}\)
Exercise \(\PageIndex{16}\)
The mass of an amoeba is about 0.000004 gram.
Exercise \(\PageIndex{17}\)
Cells in the human liver have masses of about 0.000000008 gram.
- Answer
-
\(8 \times 10^{-9}\)
Exercise \(\PageIndex{18}\)
The human sperm cell has a mass of about 0.000000000017 gram.
Exercise \(\PageIndex{19}\)
The principal protein of muscle is myosin. Myosin has a mass of 0.00000000000000000103 gram.
- Answer
-
\(1.03 \times 10^{-18}\)
Exercise \(\PageIndex{20}\)
Amino acids are molecules that combine to make up protein molecules. The amino acid tryptophan has a mass of 0.000000000000000000000340 gram.
Exercise \(\PageIndex{21}\)
An atom of the chemical element bromine has 35 electrons. The mass of a bromine atom is 0.000000000000000000000000031 gram.
- Answer
-
\(3.1 \times 10^{-26}\)
Exercise \(\PageIndex{22}\)
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 \(\PageIndex{23}\)
The approximate time it takes for a human being to die of asphyxiation is 316 seconds.
- Answer
-
\(3.16 \times 10^2\)
Exercise \(\PageIndex{24}\)
On the average, the male housefly lives 1,468,800 seconds (17 days).
Exercise \(\PageIndex{25}\)
Aluminum-26 has a half-life of 740,000 years.
- Answer
-
\(7.4 \times 10^5\)
Exercise \(\PageIndex{26}\)
Manganese-53 has a half-life of 59,918,000,000,000 seconds (1,900,000 years).
Exercise \(\PageIndex{27}\)
In its orbit around the sun, the earth moves a distance one and one half feet in about 0.0000316 second.
- Answer
-
\(3.16 \times 10^{-5}\)
Exercise \(\PageIndex{28}\)
A pi-meson is a subatomic particle that has a half-life of about 0.0000000261 second.
Exercise \(\PageIndex{29}\)
A subatomic particle called a neutral pion has a half-life of about 0.0000000000000001 second.
- Answer
-
\(1 \times 10^{-16}\)
Exercise \(\PageIndex{30}\)
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 \(\PageIndex{31}\)
The sun is about \(1 \times 10^8\) meteres from earth.
- Answer
-
100,000,000
Exercise \(\PageIndex{32}\)
The mass of the earth is about \(5.98 \times 10^{27}\) grams.
Exercise \(\PageIndex{33}\)
Light travels about \(5.866 \times 10^{12}\) miles in one year.
- Answer
-
5,866,000,000,000
Exercise \(\PageIndex{34}\)
One year is about \(3 \times 10^7\) seconds.
Exercise \(\PageIndex{35}\)
Rubik’s cube has about \(4.3 \times 10^{19}\) different configurations.
- Answer
-
43,000,000,000,000,000,000
Exercise \(\PageIndex{36}\)
A photon is a particle of light. A 100-watt light bulb emits \(1 \times 10^{20}\) photons every second.
Exercise \(\PageIndex{37}\)
There are about \(6 \times 10^7\) cells in the retina of the human eye.
- Answer
-
60,000,000
Exercise \(\PageIndex{38}\)
A car traveling at an average speed will travel a distance about equal to the length of the smallest fingernail in \(3.16 \times 10^{-4}\) seconds.
Exercise \(\PageIndex{39}\)
A ribosome of E. coli has a mass of about \(4.7 \times 10^{-19}\) grams.
- Answer
-
0.00000000000000000047
Exercise \(\PageIndex{40}\)
A mitochondrion is the energy-producing element of a cell. A mitochondrion is about \(1.5 \times 10^{-6}\) meters in diameter.
Exercise \(\PageIndex{41}\)
There is a species of frogs in Cuba that attain a length of at most \(1.25 \times 10^{-2}\) meters.
- Answer
-
0.0125
Perform the following operations.
Exercise \(\PageIndex{42}\)
\((2 \times 10^4)(3 \times 10^5)\)
Exercise \(\PageIndex{43}\)
\((4 \times 10^2)(8 \times 10^6)\)
- Answer
-
\(3.2 \times 10^9\)
Exercise \(\PageIndex{44}\)
\((6 \times 10^{14})(6 \times 10^{-10})\)
Exercise \(\PageIndex{45}\)
\((3 \times 10^{-5})(8 \times 10^7)\)
- Answer
-
\(2.4 \times 10^3\)
Exercise \(\PageIndex{46}\)
\((2 \times 10^{-1})(3 \times 10^{-5})\)
Exercise \(\PageIndex{47}\)
\((9 \times 10^{-5})(1 \times 10^{-11})\)
- Answer
-
\(9 \times 10^{-16}\)
Exercise \(\PageIndex{48}\)
\((3.1 \times 10^4)(3.1 \times 10^{-6})\)
Exercise \(\PageIndex{49}\)
\(4.2 \times 10^{-12})(3.6 \times 10^{-20})\)
- Answer
-
\(1.512 \times 10^{-31}\)
Exercise \(\PageIndex{50}\)
\((1.1 \times 10^6)^2\)
Exercises for Review
Exercise \(\PageIndex{51}\)
What integers can replace \(x\) so that the statement \(-6 < x < -2\) is true?
- Answer
-
\(-5, -4, -3\)
Exercise \(\PageIndex{52}\)
Simplify \((5x^2y^4)(2xy^5)\)
Exercise \(\PageIndex{53}\)
Determine the value of \(-[-(-|-5|)]\).
- Answer
-
\(-5\)
Exercise \(\PageIndex{54}\)
Write \(\dfrac{x^3y^{-5}}{z^{-4}\) so that only positive exponents appear.
Exercise \(\PageIndex{55}\)
Write \((2z + 1)^3(2z + 1)^{-5}\) so that only positive exponents appear.
- Answer
-
\(\dfrac{1}{(2z+1)^2}\)
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