Scientific Notation
Learning Outcomes
- Convert standard notation to scientific notation.
- Convert from scientific to standard notation.
- Apply scientific notation in an application.
A General Note: Scientific Notation
A number is written in scientific notation if it is written in the form [latex]a\times {10}^{n}[/latex], where [latex]1\le |a|<10[/latex] and [latex]n[/latex] is an integer.Example: Converting Standard Notation to Scientific Notation
Write each number in scientific notation.- Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m
- Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m
- Number of stars in Andromeda Galaxy: 1,000,000,000,000
- Diameter of electron: 0.00000000000094 m
- Probability of being struck by lightning in any single year: 0.00000143
Answer:
1. [latex]\begin{align}&\underset{\leftarrow 22\text{ places}}{{24,000,000,000,000,000,000,000\text{ m}}} \\ &2.4\times {10}^{22}\text{ m} \\ \text{ } \end{align}[/latex]
2. [latex]\begin{align}&\underset{\leftarrow 21\text{ places}}{{1,300,000,000,000,000,000,000\text{ m}}} \\ &1.3\times {10}^{21}\text{ m} \\ &\text{ } \end{align}[/latex]
3. [latex]\begin{align}&\underset{\leftarrow 12\text{ places}}{{1,000,000,000,000}} \\ &1\times {10}^{12} \\ \text{ }\end{align}[/latex]
4. [latex]\begin{align}&\underset{\rightarrow 6\text{ places}}{{0.00000000000094\text{ m}}} \\ &9.4\times {10}^{-13}\text{ m} \\ \text{ }\end{align}[/latex]
5. [latex]\begin{align}\underset{\to 6\text{ places}}{{0.00000143}} \\ 1.43\times {10}^{-6} \\ \text{ }\end{align}[/latex]
Analysis of the Solution
Observe that, if the given number is greater than 1, as in examples a–c, the exponent of 10 is positive; and if the number is less than 1, as in examples d–e, the exponent is negative.Try It
Write each number in scientific notation.- U.S. national debt per taxpayer (April 2014): $152,000
- World population (April 2014): 7,158,000,000
- World gross national income (April 2014): $85,500,000,000,000
- Time for light to travel 1 m: 0.00000000334 s
- Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715
Answer:
- [latex]$1.52\times {10}^{5}[/latex]
- [latex]7.158\times {10}^{9}[/latex]
- [latex]$8.55\times {10}^{13}[/latex]
- [latex]3.34\times {10}^{-9}[/latex]
- [latex]7.15\times {10}^{-8}[/latex]
Converting from Scientific to Standard Notation
To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal [latex]n[/latex] places to the right if [latex]n[/latex] is positive or [latex]n[/latex] places to the left if [latex]n[/latex] is negative and add zeros as needed. Remember, if [latex]n[/latex] is positive, the value of the number is greater than 1, and if [latex]n[/latex] is negative, the value of the number is less than one.Example: Converting Scientific Notation to Standard Notation
Convert each number in scientific notation to standard notation.- [latex]3.547\times {10}^{14}[/latex]
- [latex]-2\times {10}^{6}[/latex]
- [latex]7.91\times {10}^{-7}[/latex]
- [latex]-8.05\times {10}^{-12}[/latex]
Answer: 1. [latex-display]\begin{align}&3.547\times {10}^{14} \\ &\underset{\to 14\text{ places}}{{3.54700000000000}} \\ &354,700,000,000,000 \\ \text{ }\end{align}[/latex-display] 2. [latex-display]\begin{align}&-2\times {10}^{6} \\ &\underset{\to 6\text{ places}}{{-2.000000}} \\ &-2,000,000 \\ \text{ }\end{align}[/latex-display] 3. [latex-display]\begin{align}&7.91\times {10}^{-7} \\ &\underset{\to 7\text{ places}}{{0000007.91}} \\ &0.000000791 \\ \text{ }\end{align}[/latex-display] 4. [latex-display]\begin{align}&-8.05\times {10}^{-12} \\ &\underset{\to 12\text{ places}}{{-000000000008.05}} \\ &-0.00000000000805 \\ \text{ }\end{align}[/latex-display]
Try It
Convert each number in scientific notation to standard notation.- [latex]7.03\times {10}^{5}[/latex]
- [latex]-8.16\times {10}^{11}[/latex]
- [latex]-3.9\times {10}^{-13}[/latex]
- [latex]8\times {10}^{-6}[/latex]
Answer:
- [latex]703,000[/latex]
- [latex]-816,000,000,000[/latex]
- [latex]-0.00000000000039[/latex]
- [latex]0.000008[/latex]
Using Scientific Notation in Applications
Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around [latex]1.32\times {10}^{21}[/latex] molecules of water and 1 L of water holds about [latex]1.22\times {10}^{4}[/latex] average drops. Therefore, there are approximately [latex]3\cdot \left(1.32\times {10}^{21}\right)\cdot \left(1.22\times {10}^{4}\right)\approx 4.83\times {10}^{25}[/latex] atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!What properties of numbers enable operations on scientific notation?
In the example above, [latex]3\cdot \left(1.32\times {10}^{21}\right)\cdot \left(1.22\times {10}^{4}\right)\approx 4.83\times {10}^{25}[/latex]. How are we are able to simply multiply the decimal terms and add the exponents? What properties of numbers enable this? Recall that multiplication is both commutative and associative. That means, as long as multiplication is the only operation being performed, we can move the factors around to suit our needs. Lastly, the product rule for exponents, [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex], allows us to add the exponents on the base of [latex]10[/latex]. [latex]\left(3\right)\cdot\left(1.32\times {10}^{21}\right)\cdot \left(1.22\times {10}^{4}\right)=\left(3\cdot1.32\cdot1.22\right)\times\left({10}^{4}\cdot{10}^{25}\right)\approx 4.83\times {10}^{25}[/latex].Example: Using Scientific Notation
Perform the operations and write the answer in scientific notation.- [latex]\left(8.14\times {10}^{-7}\right)\left(6.5\times {10}^{10}\right)[/latex]
- [latex]\left(4\times {10}^{5}\right)\div \left(-1.52\times {10}^{9}\right)[/latex]
- [latex]\left(2.7\times {10}^{5}\right)\left(6.04\times {10}^{13}\right)[/latex]
- [latex]\left(1.2\times {10}^{8}\right)\div \left(9.6\times {10}^{5}\right)[/latex]
- [latex]\left(3.33\times {10}^{4}\right)\left(-1.05\times {10}^{7}\right)\left(5.62\times {10}^{5}\right)[/latex]
Answer: 1. [latex-display]\begin{align}\left(8.14 \times 10^{-7}\right)\left(6.5 \times 10^{10}\right) & =\left(8.14 \times 6.5\right)\left(10^{-7} \times 10^{10}\right) && \text{Commutative and associative properties of multiplication} \\ & =\left(52.91\right)\left(10^{3}\right) && \text{Product rule of exponents} \\ & =5.291 \times 10^{4} && \text{Scientific notation} \\ \text{ } \end{align}[/latex-display] 2. [latex-display]\begin{align} \left(4\times {10}^{5}\right)\div \left(-1.52\times {10}^{9}\right)& = \left(\frac{4}{-1.52}\right)\left(\frac{{10}^{5}}{{10}^{9}}\right)&& \text{Commutative and associative properties of multiplication} \\ & = \left(-2.63\right)\left({10}^{-4}\right)&& \text{Quotient rule of exponents} \\ & = -2.63\times {10}^{-4}&& \text{Scientific notation} \\ \text{ } \end{align}[/latex-display] 3. [latex-display]\begin{align} \left(2.7\times {10}^{5}\right)\left(6.04\times {10}^{13}\right)& = \left(2.7\times 6.04\right)\left({10}^{5}\times {10}^{13}\right)&& \text{Commutative and associative properties of multiplication} \\ & = \left(16.308\right)\left({10}^{18}\right)&& \text{Product rule of exponents} \\ & = 1.6308\times {10}^{19}&& \text{Scientific notation} \\ \text{ } \end{align}[/latex-display] 4. [latex-display]\begin{align} \left(1.2\times {10}^{8}\right)\div \left(9.6\times {10}^{5}\right)& = \left(\frac{1.2}{9.6}\right)\left(\frac{{10}^{8}}{{10}^{5}}\right)&& \text{Commutative and associative properties of multiplication} \\ & = \left(0.125\right)\left({10}^{3}\right)&& \text{Quotient rule of exponents} \\ & = 1.25\times {10}^{2}&& \text{Scientific notation} \\ \text{ } \end{align}[/latex-display] 5. [latex-display]\begin{align} \left(3.33\times {10}^{4}\right)\left(-1.05\times {10}^{7}\right)\left(5.62\times {10}^{5}\right)& = \left[3.33\times \left(-1.05\right)\times 5.62\right]\left({10}^{4}\times {10}^{7}\times {10}^{5}\right) \\ & \approx \left(-19.65\right)\left({10}^{16}\right) \\ & = -1.965\times {10}^{17} \end{align}[/latex-display]
Try It
Perform the operations and write the answer in scientific notation.- [latex]\left(-7.5\times {10}^{8}\right)\left(1.13\times {10}^{-2}\right)[/latex]
- [latex]\left(1.24\times {10}^{11}\right)\div \left(1.55\times {10}^{18}\right)[/latex]
- [latex]\left(3.72\times {10}^{9}\right)\left(8\times {10}^{3}\right)[/latex]
- [latex]\left(9.933\times {10}^{23}\right)\div \left(-2.31\times {10}^{17}\right)[/latex]
- [latex]\left(-6.04\times {10}^{9}\right)\left(7.3\times {10}^{2}\right)\left(-2.81\times {10}^{2}\right)[/latex]
Answer:
- [latex]-8.475\times {10}^{6}[/latex]
- [latex]8\times {10}^{-8}[/latex]
- [latex]2.976\times {10}^{13}[/latex]
- [latex]-4.3\times {10}^{6}[/latex]
- [latex]\approx 1.24\times {10}^{15}[/latex]
Example: Applying Scientific Notation to Solve Problems
In April 2014, the population of the United States was about 308,000,000 people. The national debt was about $17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.Answer: The population was [latex]308,000,000=3.08\times {10}^{8}[/latex]. The national debt was [latex]\$ 17,547,000,000,000 \approx \$1.75 \times 10^{13}[/latex]. To find the amount of debt per citizen, divide the national debt by the number of citizens.
Try It
An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures approximately 0.000008 m long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.Answer: Number of cells: [latex]3\times {10}^{13}[/latex]; length of a cell: [latex]8\times {10}^{-6}[/latex] m; total length: [latex]2.4\times {10}^{8}[/latex] m or [latex]240,000,000[/latex] m.
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