2.3 Multiplication and Division
Scientific notation is not only compact and convenient, it also simplifies arithmetic. To multiply two numbers expressed as powers of ten, you need only multiply the numbers out front and then add the exponents. If there are no numbers out front, as in [latex]100\times 100,000[/latex], then you just add the exponents (in our notation, [latex]10^2\times 10^5 = 10^7[/latex]). When there are numbers out front, you have to multiply them, but they are much easier to deal with than numbers with many zeros in them.
Here’s an example:
[latex](3\times10^5) \times (2\times10^9) = (3\times2)\times(10^{5+9}) = 6\times10^{14}[/latex]
And here’s another example:
[latex]\begin{align*}0.04\times 6,000,000 &= (4\times10^{-2}) \times (6\times10^6)\\ &= (4\times6)\times(10^{-2+6})\\ &= 24 \times10^4\\ &= 2.4\times10^5\end{align*}[/latex]
Note in the second example that when we added the exponents, we treated negative exponents as we do in regular arithmetic (-2 plus 6 equals 4). Also, notice that our first result had a 24 in it, which was not in the acceptable form, having two places to the left of the decimal point, and we therefore changed it to 2.4 and changed the exponent accordingly.
To divide, you divide the numbers out front and subtract the exponents. Here are several examples:
- [latex]\dfrac{1,000,000}{1000} = \dfrac{10^6}{10^3} = 10^{6-3} = 10^3[/latex]
- [latex]\dfrac{9\times10^{12}}{2\times10^{3}} = 4.5\times10^{12-3} = 4.5\times10^{9}[/latex]
- [latex]\dfrac{2.8\times10^2}{6.2\times10^5} =.452\times10^{2-5} = .452\times10^{-3} = 4.52\times10^{-4}[/latex]
In the last example, our first result was not in the standard form, so we had to change 0.452 into 4.52, and change the exponent accordingly.
If this is the first time that you have met scientific notation, we urge you to practice many examples using it. You might start by solving the exercises below. Like any new language, the notation looks complicated at first but gets easier as you practice it.
Example 2.2
- At the end of September 2015, the New Horizons spacecraft (which encountered Pluto for the first time in July 2015) was [latex]4.898[/latex] billion km from Earth. Convert this number to scientific notation. How many astronomical units is this? (An astronomical unit is the distance from Earth to the Sun, or about [latex]150[/latex] million km.)
Solution
[latex]4.898[/latex] billion is [latex]4.898\times 10^9[/latex] km. One astronomical unit (AU) is [latex]150[/latex] million km = [latex]1.5\times 10^8[/latex] km. Dividing the first number by the second, we get [latex]3.27\times 10^{(9 - 8)} = 3.27\times 10^1[/latex] AU.
- During the first six years of its operation, the Hubble Space Telescope circled Earth [latex]37,000[/latex] times, for a total of [latex]1,280,000,000[/latex] km. Use scientific notation to find the number of km in one orbit.
Solution
[latex]\begin{align*}\frac{(1.28\times 10^9\;\text{km})}{ (3.7\times 10^4\;\text{orbits})}=0.346\times 10^{(9-4)}=0.346\times 10^5= 3.46\times 10^4\;\text{km per orbit}\end{align*}[/latex]
- In a large university cafeteria, a soybean-vegetable burger is offered as an alternative to regular hamburgers. If [latex]889,875[/latex] burgers were eaten during the course of a school year, and [latex]997[/latex] of them were veggie-burgers, what fraction and what percent of the burgers does this represent?
Solution
[latex]\begin{align*}\frac{(9.97\times 10^2\;\text{veggie burgers})}{ (8.90\times 10^5\;\text{total burgers})}=1.12\times 10^{(2-5)}= 1.12\times 10^{-3}\end{align*}[/latex]
(or roughly about one thousandth) of the burgers were vegetarian. Percent means per hundred. So
[latex]1.12\times 10^{-3}\times 10^2 = 1.12\times 10^{(-3+2)}=1.12\times 10^{-1}[/latex] percent
- In a 2012 Kelton Research poll, [latex]36[/latex] percent of adult Americans thought that alien beings have actually landed on Earth. The number of adults in the United States in 2012 was about [latex]222,000,000[/latex]. Use scientific notation to determine how many adults believe aliens have visited Earth.
Solution
[latex]36\%[/latex] is [latex]36[/latex] hundredths or [latex]0.36[/latex] or [latex]3.6\times 10^{-1}[/latex]. Multiply that by [latex]2.22\times 10^8[/latex] and you get about [latex]7.99\times 10^{(-1+8)} = 7.99\times 10^7[/latex] or almost [latex]80[/latex] million people who believe that aliens have landed on our planet. We need more astronomy courses to educate all those people.
- In the school year 2009–2010, American colleges and universities awarded [latex]2,354,678[/latex] degrees. Among these were [latex]48,069[/latex] PhD degrees. What fraction of the degrees were PhDs? Express this number as a percent. (Now go and find a job for all those PhDs!)
Solution
[latex]\begin{align*}\frac{(4.81\times 10^4)}{(2.35\times 10^6)}=2.05\times 10^{(4 - 6)}= 2.05\times 10^{-2}=\text{about}\;2\%\end{align*}[/latex]
(Note that in these examples we are rounding off some of the numbers so that we don’t have more than [latex]2[/latex] places after the decimal point.)
- A star [latex]60[/latex] light-years away has been found to have a large planet orbiting it. Your uncle wants to know the distance to this planet in old-fashioned miles. Assume light travels [latex]186,000[/latex] miles per second, and there are [latex]60[/latex] seconds in a minute, [latex]60[/latex] minutes in an hour, [latex]24[/latex] hours in a day, and [latex]365[/latex] days in a year. How many miles away is that star?
Solution
One light-year is the distance that light travels in one year. (Usually, we use metric units and not the old British system that the United States is still using, but we are going to humor your uncle and stick with miles.) If light travels [latex]186,000[/latex] miles every second, then it will travel [latex]60[/latex] times that in a minute, and [latex]60[/latex] times that in an hour, and [latex]24[/latex] times that in a day, and [latex]365[/latex] times that in a year. So we have
[latex]1.86\times 10^5\times 6.0\times 10^1\times 6.0\times 10^1\times 2.4\times 10^1\times 3.65\times 10^2[/latex].
So we multiply all the numbers out front together and add all the exponents. We get [latex]586.57\times 10^{10} = 5.86\times 10^{12}[/latex] miles in a light year (which is roughly [latex]6[/latex] trillion miles—a heck of a lot of miles). So if the star is [latex]60[/latex] light-years away, its distance in miles is
[latex]6\times 10^1\times 5.86\times 10^{12} = 35.16\times 10^{13} = 3.516\times 10^{14}[/latex] miles.
Attribution
“A.3 Scientific Notation” from Douglas College Astronomy 1105 by Douglas College Department of Physics and Astronomy, is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. Adapted from Astronomy 2e.
