0.3 Scientific notation  (Page 2/2)

Writing small numbers

Now take a number like 0.00347, which is also not in the standard (agreed-to) form for scientific notation. To put it into that format, we must make the first part of it 3.47 by moving the decimal point three places to the right . Note that this motion to the right is the opposite of the motion to the left that we discussed above. To keep track, we call this change negative and put a minus sign in the exponent. Thus 0.00347 becomes 3.47 × 10 ⁻³ .

In the example we gave at the beginning, the mass of the hydrogen atom would then be written as 1.67 × 10 ⁻²⁷ kg. In this system, one is written as 10 ⁰ , a tenth as 10 ⁻¹ , a hundredth as 10 ⁻² , and so on. Note that any number, no matter how large or how small, can be expressed in scientific notation.

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 100 × 100,000, then you just add the exponents (in our notation, 10 ² × 10 ⁵ = 10 ⁷ ). 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:

And here’s another example:

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:

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.

Exercises

1. At the end of September, 2015, the New Horizons spacecraft (which encountered Pluto for the first time in July 2015) was 4.898 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 150 million km.)
2. During the first six years of its operation, the Hubble Space Telescope circled Earth 37,000 times, for a total of 1,280,000,000 km. Use scientific notation to find the number of km in one orbit.
3. In a large university cafeteria, a soybean-vegetable burger is offered as an alternative to regular hamburgers. If 889,875 burgers were eaten during the course of a school year, and 997 of them were veggie-burgers, what fraction and what percent of the burgers does this represent?
4. In a 2012 Kelton Research poll, 36 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 222,000,000. Use scientific notation to determine how many adults believe aliens have visited Earth.
5. In the school year 2009–2010, American colleges and universities awarded 2,354,678 degrees. Among these were 48,069 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!)
6. A star 60 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 186,000 miles per second, and there are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 365 days in a year. How many miles away is that star?

Answers

1. 4.898 billion is 4.898 × 10 ⁹ km. One astronomical unit (AU) is 150 million km = 1.5 × 10 ⁸ km. Dividing the first number by the second, we get 3.27 × 10 ^{(9 – 8)} = 3.27 × 10 ¹ AU.
2. $\frac{1.28\times {10}^9\text{km}}{3.7\times {10}^4\text{orbits}}=0.346\times {10}^{(9-4)}=0.346\times {10}^5=3.46\times {10}^4\text{km per orbit}.$
3. $\frac{9.97\times {10}^2\text{veggie burgers}}{8.90\times {10}^5\text{total burgers}}=1.12\times {10}^{(2-5)}=1.12\times {10}^{(2-5)}=1.12\times {10}^{\mathrm{-3}}$ (or roughly about one thousandth) of the burgers were vegetarian. Percent means per hundred. So $\frac{1.12\times {10}^{\mathrm{-3}}}{{10}^{\mathrm{-2}}}=1.12\times {10}^{(\mathrm{-3}-(\mathrm{-2}))}=1.12\times {10}^{\mathrm{-1}}\text{percent}$ (which is roughly one tenth of one percent).
4. 36% is 36 hundredths or 0.36 or 3.6 × 10 ⁻¹ . Multiply that by 2.22 × 10 ⁸ and you get about 7.99 × 10 ^{(−1 + 8)} = 7.99 × 10 ⁷ or almost 80 million people who believe that aliens have landed on our planet. We need more astronomy courses to educate all those people.
5. $\frac{4.81\times {10}^4}{2.35\times {10}^6}=2.05\times {10}^{(4-6)}=2.05\times {10}^{\mathrm{-2}}=\text{about}2\text{\%}$ . (Note that in these examples we are rounding off some of the numbers so that we don’t have more than 2 places after the decimal point.)
6. 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 186,000 miles every second, then it will travel 60 times that in a minute, and 60 times that in an hour, and 24 times that in a day, and 365 times that in a year. So we have 1.86 × 10 ⁵ × 6.0 × 10 ¹ × 6.0 × 10 ¹ × 2.4 × 10 ¹ × 3.65 × 10 ² . So we multiply all the numbers out front together and add all the exponents. We get 586.57 × 10 ¹⁰ = 5.86 × 10 ¹² miles in a light year (which is roughly 6 trillion miles—a heck of a lot of miles). So if the star is 60 light-years away, its distance in miles is 6 × 10 ¹ × 5.86 × 10 ¹² = 35.16 × 10 ¹³ = 3.516 × 10 ¹⁴ miles.