8.4: Scientific Notation
Sometimes in mathematics numbers are huge. They are so huge that we use what is called scientific notation. It is easier to work with such numbers when we shorten their decimal places and multiply them by 10 to a specific power. In this lesson, you will learn how to use scientific notation by hand and on a calculator.
Powers of 10:
\begin{align*}100,000 &= 10^5\\ 10,000 &= 10^4\\ 1,000 &= 10^3\\ 100 &= 10^2\\ 10 &= 10^1\end{align*}
Using Scientific Notation for Large Numbers
Example: If we divide 643,297 by 100,000 we get 6.43297. If we multiply this quotient by 100,000, we get back to our original number. But we have just seen that 100,000 is the same as \begin{align*}10^5\end{align*}, so if we multiply 6.43297 by \begin{align*}10^5\end{align*}, we should also get our original answer. In other words \begin{align*}6.43297 \times 10^5=643,297\end{align*} Because there are five zeros, the decimal moves over five places.
Solution: Look at the following examples:
\begin{align*}2.08 \times 10^4 &= 20,800\\ 2.08 \times 10^3 &= 2,080\\ 2.08 \times 10^2 &= 208\\ 2.08 \times 10^1 &= 20.8\\ 2.08 \times 10^0 &= 2.08\end{align*}
The power tells how many decimal places to move; positive powers mean the decimal moves to the right. A positive 4 means the decimal moves four positions the right.
Example 1: Write in scientific notation.
653,937,000
Solution: \begin{align*}653,937,000=6.53937000 \times 100,000,000=6.53937 \times 10^8\end{align*}
Oftentimes we do not keep more than a few decimal places when using scientific notation and we round the number to the nearest whole number, tenth, or hundredth depending on what the directions say. Rounding Example 1 could look like \begin{align*}6.5 \times 10^8\end{align*}.
Using Scientific Notation for Small Numbers
We’ve seen that scientific notation is useful when dealing with large numbers. It is also good to use when dealing with extremely small numbers.
Look at the following examples:
\begin{align*}2.08 \times 10^{-1} &= 0.208\\ 2.08 \times 10^{-2} &= 0.0208\\ 2.08 \times 10^{-3} &= 0.00208\\ 2.08 \times 10^{-4} &= 0.000208\end{align*}
Example 2: The time taken for a light beam to cross a football pitch is 0.0000004 seconds. Write in scientific notation.
Solution: \begin{align*}0.0000004=4 \times 0.0000001=4 \times \frac{1}{10,000,000}=4 \times \frac{1}{10^7}=4 \times 10^{-7}\end{align*}
Evaluating Expressions Using Scientific Notation
When evaluating expressions with scientific notation, it is easiest to keep the powers of 10 together and deal with them separately.
Example: \begin{align*}(3.2 \times 10^6) \cdot (8.7 \times 10^{11}) = 3.2 \times 8.7 \cdot 10^6 \times 10^{11} = 27.84 \times 10^{17}=2.784 \times 10^1 \times 10^{17} = 2.784 \times 10^{18}\end{align*}
Solution: It is best to keep one number before the decimal point. In order to do that, we had to make 27.84 become \begin{align*}2.784 \times 10^1\end{align*} so we could evaluate the expression more simply.
Example 3: Evaluate the following expression.
(a) \begin{align*}(1.7 \times 10^6) \cdot (2.7 \times 10^{-11})\end{align*}
(b) \begin{align*}(3.2 \times 10^6) \div (8.7 \times 10^{11})\end{align*}
Solution:
(a) \begin{align*}(1.7 \times 10^6) \cdot (2.7 \times 10^{-11})=1.7 \times 2.7 \cdot 10^6 \times 10^{-11}=4.59 \times 10^{-5}\end{align*}
(b) \begin{align*}(3.2 \times 10^6) \div (8.7 \times 10^{11})=\frac{3.2 \times 10^6}{8.7 \times 10^{11}}=\frac{3.2}{8.7} \times \frac{10^6}{10^{11}}=0.368 \times 10^{6-11} = 3.68 \times 10^{-1} \times 10^{-5}=3.68 \times 10^{-6}\end{align*}
You must remember to keep the powers of ten together, and have 1 number before the decimal.
Scientific Notation Using a Calculator
Scientific and graphing calculators make scientific notation easier. To compute scientific notation, use the [EE] button. This is [2nd] [,] on some TI models or \begin{align*}[10^\chi]\end{align*}, which is [2nd] [log].
For example to enter \begin{align*}2.6 \times 10^5\end{align*} enter 2.6 [EE] 5.
When you hit [ENTER] the calculator displays \begin{align*}260000\end{align*} if it’s set in Scientific mode OR it displays 260,000 if it’s set in Normal mode.
Solving Real-World Problems Using Scientific Notation
Example: The mass of a single lithium atom is approximately one percent of one millionth of one billionth of one billionth of one kilogram. Express this mass in scientific notation.
Solution: We know that percent means we divide by 100, and so our calculation for the mass (in kg) is \begin{align*}\frac{1}{100} \times \frac{1}{1,000,000} \times \frac{1}{1,000,000,000} \times \frac{1}{1,000,000,000} = 10^{-2} \times 10^{-6} \times 10^{-9} \times 10^{-9}\end{align*}
Next, we use the product of powers rule we learned earlier in the chapter.
\begin{align*}10^{-2} \times 10^{-6} \times 10^{-9} \times 10^{-9}=10^{((-2)+(-6)+(-9)+(-9))}=10^{-26} \ kg.\end{align*}
The mass of one lithium atom is approximately \begin{align*}1 \times 10^{-26} \ kg\end{align*}.
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK-12 Basic Algebra: Scientific Notation (14:26)
Write the numerical value of the following expressions.
- \begin{align*}3.102 \times 10^2\end{align*}
- \begin{align*}7.4 \times 10^4\end{align*}
- \begin{align*}1.75 \times 10^{-3}\end{align*}
- \begin{align*}2.9 \times 10^{-5}\end{align*}
- \begin{align*}9.99 \times 10^{-9}\end{align*}
- \begin{align*}(3.2 \times 10^6) \cdot (8.7 \times 10^{11})\end{align*}
- \begin{align*}(5.2 \times 10^{-4}) \cdot (3.8 \times 10^{-19})\end{align*}
- \begin{align*}(1.7 \times 10^6) \cdot (2.7 \times 10^{-11})\end{align*}
- \begin{align*}(3.2 \times 10^6) \div (8.7 \times 10^{11})\end{align*}
- \begin{align*}(5.2 \times 10^{-4}) \div (3.8 \times 10^{-19})\end{align*}
- \begin{align*}(1.7 \times 10^6) \div (2.7 \times 10^{-11})\end{align*}
Write the following numbers in scientific notation.
- 120,000
- 1,765,244
- 63
- 9,654
- 653,937,000
- 1,000,000,006
- 12
- 0.00281
- 0.000000027
- 0.003
- 0.000056
- 0.00005007
- 0.00000000000954
- The moon is approximately a sphere with radius \begin{align*}r=1.08 \times 10^3\end{align*} miles. Use the formula Surface \begin{align*}\text{Area}=4 \pi r^2\end{align*} to determine the surface area of the moon, in square miles. Express your answer in scientific notation, rounded to two significant figures.
- The charge on one electron is approximately \begin{align*}1.60 \times 10^{-19}\end{align*} coulombs. One Faraday is equal to the total charge on \begin{align*}6.02 \times 10^{23}\end{align*} electrons. What, in coulombs, is the charge on one Faraday?
- Proxima Centauri, the next closest star to our Sun, is approximately \begin{align*}2.5 \times 10^{13}\end{align*} miles away. If light from Proxima Centauri takes \begin{align*}3.7 \times 10^4\end{align*} hours to reach us from there, calculate the speed of light in miles per hour. Express your answer in scientific notation, rounded to two significant figures.
Mixed Review
- 14 milliliters of a 40% sugar solution was mixed with 4 milliliters of pure water. What is the concentration of the mixture?
- Solve the system \begin{align*}\begin{cases} 6x+3y+18\\ -15=11y-5x \end{cases}\end{align*}.
- Graph the function by creating a table: \begin{align*}f(x)=2x^2\end{align*}. Use the following values for \begin{align*}x: -5 \le x \le 5\end{align*}.
- Simplify \begin{align*}\frac{5a^6 b^2 c^{-6}}{a^{11} b}\end{align*}. Your answer should have only positive exponents.
- Each year Americans produce about 230 million tons of trash (Source: http://www.learner.org/interactives/garbage/solidwaste.html). There are 307,006,550 people in the United States. How much trash is produced per person per year?
- The volume of a 3-dimesional box is given by the formula: \begin{align*}V=l(w)(h)\end{align*}, where \begin{align*}l=\end{align*} length, \begin{align*}w=\end{align*} width, and \begin{align*}h=\end{align*} height of the box. The box holds 312 cubic inches and has a length of 12 inches and a width of 8 inches. How tall is the box?
Quick Quiz
- Simplify: \begin{align*}\frac{(2x^{-4}y^3)^{-3} \ \cdot \ x^{-3} y^{-2}}{-2x^0y^2}\end{align*}.
- The formula \begin{align*}A=1,500(1.0025)^t\end{align*} gives the total amount of money in a bank account with a balance of $1,500.00, earning 0.25% interest, compounded annually. How much money would be in the account five years in the past?
- True or false? \begin{align*}\left(\frac{5}{4}\right)^{-3}= -\frac{125}{64}\end{align*}