Metric prefixes and scientific notation

Section 1.5 Metric prefixes and scientific notation

In this section we’ll learn about how something called scientific notation helps us to more efficiently handle very large and very small numbers

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Opening Activity 1.5.1.

In everyday language, we use number names like “thousand,”, “million,” “billion,” and maybe even “trillion” to describe very large numbers:

\begin{align*} \amp 1 \textbf{ thousand} \amp \amp =\amp 1{,}000\amp\\ \amp 1 \textbf{ million} \amp \amp =\amp 1{,}000{,}000\amp\\ \amp 1 \textbf{ billion} \amp \amp=\amp 1{,}000{,}000{,}000\amp\\ \amp 1 \textbf{ trillion} \amp \amp=\amp 1{,}000{,}000{,}000{,}000\amp \end{align*}

Let’s spend some time thinking about how these numbers relate to each other. For instance, since

\begin{equation*} \dfrac{1 \textbf{ million}}{1 \textbf{ thousand}} = \dfrac{1{,}000{,}\cancel{000}}{1{,}\cancel{000}} = 1{,}000\text{,} \end{equation*}

a million is a thousand thousands.

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How many millions is one billion?
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How many billions is one trillion?
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How many thousands is one million?
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Subsection Tara’s files

Tara is working on a big project at work. She wants to back up her files to her online drop box. The site says she has 72 GB of memory remaining. Tara has about 200 files at an average of 42.3 MB each that she would like to upload. Will she have room?

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To answer Tara’s question we need to know that GB is short for “gigabyte” and MB is short for “megabyte.” A byte is a very small unit of computer memory storage space just enough for about one letter. You may have heard the word “mega” used to mean “really big.” There’s a reason for that. Mega is short for 1 million. That’s pretty big. But giga stands for 1 billion, so that’s even bigger — a thousand times bigger!

\begin{align*} \amp\textbf{mega} \amp \amp= \amp \amp 1 \textbf{ million} \amp \amp =\amp 1{,}000{,}000\amp\\ \amp\textbf{giga} \amp \amp=\amp \amp1 \textbf{ billion} \amp \amp=\amp 1{,}000{,}000{,}000\amp \end{align*}

What all this means is Tara has

\begin{equation*} 72\text{ GB} = 72 \text{ billion bytes} = 72{,}000{,}000{,}000 \text{ bytes} \end{equation*}

of memory remaining. She would like to save 200 files at 42.3 MB each which comes to

\begin{equation*} 200 \times 42.3 = 8{,}460 \text{ MB} \end{equation*}

which is really

\begin{equation*} 8{,}460 \text{ MB} = 8{,}460\text{ million bytes} = 8{,}460{,}000{,}000 \text{ bytes} \end{equation*}

Finding it hard to compare all those zeros? Try this.

\begin{equation*} 8{,}460\text{ MB} = 8{,}460{,}000{,}000\text{ bytes} = 8.46 \text{ GB } < 9 \text{ GB} \end{equation*}

So Tara wants to store less than 9 GB of information and she has 72 GB remaining. She has plenty of room. Save away.

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Activity 1.5.2.

In 2011, the population of the United States was approximately 0.313 billion.

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Use the fact that 1 billion = 1,000 million to write down two unit conversion fractions:

\begin{equation*} 1 = \dfrac{\fillinmath{1.000} \text{ billion}} {\fillinmath{1000} \text{ million}} \quad \text{and} \quad 1 = \dfrac{\fillinmath{1000} \text{ million}} {\fillinmath{1.000} \text{ billion}} \end{equation*}

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Writing the population as 0.313 billion seems strange. A more natural unit would be millions. Use the correct unit conversion fraction from above to rewrite the population in millions of people.
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Subsection Downloading photos

Tara also needs to download about 700 MB of rather high quality photos. Her computer downloads photos at 187 kbps. How long will it take? (And does she have time to run for a coffee?) The mysterious kbps stands for kilobits (Kb) per second. Like mega and giga, the word “kilo” stands for a large number, in this case 1,000.

\begin{equation*} \textbf{kilo} = 1\textbf{ thousand} = 1{,}000 \end{equation*}

That’s the same word “kilo” as in kilometer (about mile) or kilogram (about pound) and there’s good reason for that as

\begin{align*} \textbf{kilometer} \amp= 1{,}000 \text{ meters}\\ \textbf{kilogram} \amp= 1{,}000 \text{ grams} \end{align*}

Perhaps you’ve seen the letter K as short for a thousand? That’s where it comes from.

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Aside

Back to Tara. Her download speed is 187 kilobits per second. Perhaps this is the right moment to mention that a bit is even smaller than a byte.

\begin{equation*} 1 \text{ byte} = 8 \text{ bits} \end{equation*}

How long will it take Tara to download 700 MB? We can think of this calculation as a unit conversion by imagining

\begin{equation*} 187 \text{ kilobits} = 1 \text{ second} \end{equation*}

Watch.

\begin{equation*} 700 \cancel{\text{ MB}} \ast \frac{1{,}000{,}000 \cancel{\text{ bytes}}} {1 \cancel{\text{ MB}}} \ast \frac{8 \cancel{\text{ bits}}} {1 \cancel{\text{ byte}}} \ast \frac{1 \cancel{\text{ kilobit}}} {1{,}000 \cancel{\text{ bits}}} \ast \frac{1 \text{ second}} {187 \cancel{\text{ kilobits}}} \end{equation*}

\begin{equation*} = 700 \times 1{,}000{,}000 \times 8 \div 1{,}000 \div 187 = 29{,}946.524\ldots\text{ seconds} \end{equation*}

Let’s convert to a more reasonable unit.

\begin{equation*} 29{,}946.524\ldots \cancel{\text{ seconds}} \ast \frac{1 \cancel{\text{ minute}}} {60 \cancel{\text{ seconds}}} \ast \frac{1 \text{ hour}} {60 \cancel{\text{ minutes}}} \end{equation*}

\begin{equation*} = 29{,}946.524\ldots \div 60 \div 60 = 8.318\ldots \approx 8.32 \text{ hours} \end{equation*}

It will take Tara over 8 hours to download those photos. Perhaps Tara should compress the photos into a zip file or use lower resolution or find a way to download faster. Or, she can just set it up to download overnight.

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Activity 1.5.3.

Tara’s coworker Brandon has a much faster Internet connection on his computer at 1,500 kbps.

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How long would it take Brandon to download 700 MB?
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How much information could Brandon download in 8 hours?
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Subsection The metric system and powers of ten

What do Tara’s two problems have in common? Both of them involve the metric system. The metric system of measurement, or international system of units (ISU), is the official system of nearly all countries, the United States being a notable exception. Science, international trade, and most international sporting events like the Olympics are based in the metric system. In the United States system (known officially at the British system or, since the British stopped using it, the imperial system of measurement), we have all sorts of difficult to remember conversions.

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One notable feature of the metric system is that most units come in sizes ranging from small to large: the (metric) prefixes like kilo, mega, or giga tell us which size.

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Really large numbers, like 8,460,000,000, are awkward to read and awkward to work with. We have seen how metric prefixes allow us to rewrite these large numbers in a way that’s much easier both to read and to work with. There’s another option that’s used often in the sciences (and by your calculator). To explain it we need to understand powers of 10.

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Perhaps you know what happens when we multiply a number by 10, like $5 \times 10 = 50$ or, more appropriate to our example,

\begin{equation*} 8.46 \times 10 = 84.6 \end{equation*}

The effect of multiplying by 10 is to move the decimal point one place to the right. When we multiply by 1,000 we get $5 \times 1{,}000 = 5{,}000$ or, for our example,

\begin{equation*} 8.46 \times 1{,}000 = 8{,}460 \end{equation*}

The effect of multiplying by 1,000 is to move the decimal point three places to the right. The connection is that

\begin{equation*} 10 \times 10 \times 10 =1{,}000 \end{equation*}

Each $\times 10$ has the effect of moving the decimal point one place to the right so $\times 1{,}000$ has the same effect as multiplying by 10 three times, so the decimal point moves three places to the right. That means

\begin{align*} 8{,}460{,}000{,}000 \amp = 8.46 \underbrace{\hbox{$\times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 $}}_{\hbox{9 times}} \\ \amp = 8.46 \ast 10^ 9 \end{align*}

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Since we’re multiplying by the same number (10) over and over again, it’s easier to use exponential notation. Here 10 is the base and 9 is the exponent (or power). In this context, the exponent is also called the order of magnitude.

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The point of this calculation was that

\begin{equation*} 8{,}460{,}000{,}000 = 8.46\ast10^9 \end{equation*}

This shorthand is called scientific notation. The base is always 10. The exponent is always a whole number. The number out front, like 8.46 in our example, is always between 1 and 10, which means there’s exactly one digit before the decimal point (and any others must come afterwards). It is customary to use $\times$ instead of $\ast$ in scientific notation, so we should write

\begin{equation*} 8{,}460{,}000{,}000 = 8.46\times10^9 \end{equation*}

As another example, we saw earlier that

\begin{equation*} 5{,}000 = 5 \times 1{,}000 = 5 \times 10^3 \end{equation*}

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Most calculators use the $\wedge$ symbol for exponents, as do most computer software packages. Two other notations calculators sometimes use are $y^x$ or $x^y\text{.}$ Sometimes that operation is accessible through the 2nd or shift key; something like SHIFT $\times\text{.}$ If you’re not sure, ask a classmate or your instructor. For practice, check that

\begin{equation*} 5 \times 10^3 = 5 \times 10 \wedge 3 = 5{,}000 \quad \checkmark \end{equation*}

Notice that the order of operations is exactly what we wanted here: $5 \times 10 \wedge 3$ first raises 10 to the 3rd power and then multiplies by 5. So we can enter it all at once without needing parentheses.

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Here’s the full list of the order of operations, the priority ranking for arithmetic operations.

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Order of Operations

First, calculate anything inside Parentheses ().
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Next, calculate Exponents $\wedge\text{,}$ in order from left to right.
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Then, Multiply $\times$ and Divide $\div\text{,}$ in order from left to right.
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Last, Add $+$ and Subtract $-\text{,}$ in order from left to right.
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We highlighted the letters PEMDAS which often helps people remember this order. (“Please Excuse My Dear Aunt Sally” is how I learned it.) The good news is that your calculator does the operations in exactly this order. And if you want something in a different order, all you need to do is use parentheses around quantities you want calculated first.

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Back to our large number. Enter

\begin{equation*} 8.46 \times 10 \wedge 9= \end{equation*}

What do you see? Some calculators correctly list out $8{,}460{,}000{,}000$ while others report the number back in scientific notation, which is not particularly useful. (Sigh.)

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Let’s try a number so big that (nearly) every calculator will switch to scientific notation. Enter

\begin{equation*} 2.7\times 10 \wedge 30= \end{equation*}

Look carefully at the screen. Your calculator might display something like

\begin{equation*} \boxed{~2.70000000 ~ {\small \text E} ~30~} \quad\text{or}\quad \boxed{~2.70000000~ _{ \times 10}~30~} \end{equation*}

Whatever shorthand your calculator uses, you should write

\begin{equation*} 2.7 \times 10^{30} \end{equation*}

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Interested in what that number is in our usual decimal notation? It’s

\begin{equation*} 2, \underbrace{ \hbox{$700{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000 $}}_\text{decimal point moves 30 places } \end{equation*}

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Activity 1.5.4.

In 2011, the GDP (gross domestic product) of the United States was approximately $15,596 billion, and the population of the United States was approximately 0.313 billion.

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Rewrite the population as a number of people, both in normal decimal notation (that means with all the 0s) and in scientific notation.
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Rewrite the GDP in dollars, both in normal decimal notation and in scientific notation.
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Subsection Very small things

Poor Tara is pulling her hair out over this project. Well, not literally, but she is quite frustrated over how slowly the project is going. She wonders: how thick is a human hair? And, how many hairs would you need to lay out to span an inch?

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Turns out that a typical human hair is about $0.00012$ meters across. Very small numbers are also awkward to read and awkward to work with. In this section, we write $0.000~12$ where we insert a space to help you read the number. Of course, a better solution is to use metric prefixes to get more appropriate units, just as we did for large numbers.

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For example, centi is short for 1 in a hundred, or $0.01\text{.}$ Not surprising since one cent is $0.01, or one percent is 1%=0.01. That’s the same word “centi” as in centimeter (about inch) and there’s good reason for that as

\begin{equation*} 1\text{ meter} = 100 \text{ centimeters} \end{equation*}

Similarly, milli is short for 1 in a thousand and micro is short for 1 in a million.

\begin{align*} \amp\textbf{centi} \amp \amp=\amp \amp1 \text{ in a hundred} \amp \amp=\amp 0\amp.01\\ \amp\textbf{milli} \amp \amp=\amp \amp1 \text{ in a thousand} \amp \amp=\amp 0\amp.001\\ \amp\textbf{micro} \amp \amp=\amp \amp1 \text{ in a million} \amp \amp=\amp 0\amp.000~001 \end{align*}

What about that human hair? It is convenient to measure in micrometers using that

\begin{equation*} 1 \text{ meter} = 1{,}000{,}000\text{ micrometers} \end{equation*}

The width of a human hair in micrometers (abbreviated $\mu$m in the sciences) is

\begin{equation*} 0.000~12 \cancel{\text{ meters} } \ast \frac{1{,}000{,}000\, \mu \text{m}} {1 \cancel{\text{ meter}}} = 0.000~12 \times 1{,}000{,}000 = 120\, \mu \text{m} \end{equation*}

The $\mu$ symbol is the Greek letter mu, but we’ll just read $\mu m$ as micrometers.

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To answer Tara’s question about how many hairs in an inch, we recall that

\begin{equation*} 1 \text{ inch} \approx 2.54 \text{ cm} \end{equation*}

where cm is short for centimeter. Ready to convert?

\begin{equation*} 1 \cancel{\text{ inch}} \ast \frac{ 2.54 \cancel{\text{ cm}} } {1 \cancel{\text{ inch}}} \ast \frac{1 \cancel{\text{ meter}}} {100 \cancel{\text{ cm}}} \ast \frac{1{,}000{,}000 \cancel{\mu m}} {1 \cancel{\text{ meter}}} \ast \frac{1 \text{ hair}} {120 \cancel{\mu m}} \end{equation*}

\begin{equation*} = 2.54 \div 100 \times 1{,}000{,}000 \div 120 = 211.66666\ldots \approx 200 \text{ hairs} \end{equation*}

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Activity 1.5.5.

Souksavanh is setting up a patient’s intravenous (IV) medication. She sets the drip at 42 drops/minute. The drip chamber size is 20 drops/mL. Recall

\begin{align*} \amp\textbf{milli} \amp \amp= \amp \amp 1 \text{ in a thousand} \amp \amp =\amp 0\amp.001\amp\\ \amp\textbf{micro} \amp \amp= \amp \amp 1 \text{ in a million} \amp \amp =\amp 0\amp.000~001\amp \end{align*}

At what rate is the IV fluid being delivered to Souk’s patient? Answer in mL/min (millileters per minute).

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Subsection Very small numbers in scientific notation

We can also describe really small numbers using scientific notation. Perhaps you know what happens when we divide a number by 10, like $50 \div 10 = 5$ or, more appropriate to our example,

\begin{equation*} 1.2 \div 10 = 0.12 \end{equation*}

The effect of dividing by 10 is to move the decimal point one place to the left. If we divide by 1,000,000 instead, we get

\begin{equation*} 1.2 \div 1{,}000{,}000 = 0.000~001~2 \end{equation*}

The connection is that

\begin{equation*} 1{,}000{,}000 = 10 \wedge 6 \end{equation*}

and so dividing by 1,000,000 moves the decimal point six places to the left. Notice that we have to introduce zeros as placeholders.

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The width of a hair was 0.00012 meters. To get that number from 1.2, we need to move the decimal point 4 places to the left.

\begin{equation*} 1.2 \div 10^4 = 1.2 \div 10{,}000 = 0.000~12 \end{equation*}

The shorthand for dividing by a power is to use negative exponents. For example

\begin{equation*} \div 10^4 = \times 10^{-4} \end{equation*}

It has nothing to do with negative numbers. It’s just a shorthand. The point of this calculation was that

\begin{equation*} 0.00012= 1.2\ast10^{-4} \end{equation*}

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Once again we have scientific notation. The base is still 10. The exponent is still a whole number, although now it’s negative. The number out front, like 1.2 in our example, is still between 1 and 10, which means there’s exactly one digit before the decimal point (and any others must come afterwards). As before it customary to use $\times$ instead of $\ast$ in scientific notation, so we should write

\begin{equation*} 0.000~12= 1.2 \times 10^{-4} \end{equation*}

When you see a number written in scientific notation, the power of 10 tells you a lot. For example, $6.7 \times 10^4 = 67{,}000$ and $6.7 \times 10^{-3} = 0.006~7\text{.}$ A positive power of 10 says you have a big number, and a negative power of 10 says you’re dealing with a very small number.

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Activity 1.5.6.

After vacuuming his apartment, Edgar did a little research online and found out that the mass of a dust particle is

\begin{equation*} 0.000~000~000~753\text{ kilograms.} \end{equation*}

(The strange-looking spaces are to help you see that there are 9 zeros in the number.)

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Write the mass of a dust particle in kilograms, using scientific notation.
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Write the mass of a dust particle in grams, using scientific notation. (Remember that 1 kilogram = 1,000 grams.)
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Do you know …

How to calculate powers on your calculator?
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What million, billion, and trillion mean?
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Why metric prefixes are used?
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What common metric prefixes (mega, giga, kilo, centi, milli, micro, nano) mean?
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Ask your instructor which prefixes you need to remember, and whether any prefixes will be provided during the exam.
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Why scientific notation is used?
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The standard format for scientific notation?
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What kinds of numbers have a positive order of magnitude, and which have a negative order of magnitude?
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How to convert between decimal notation and scientific notation?
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How your calculator reports numbers in scientific notation, and what (might be) different when you’re reporting that number?
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The usual order of operations (PEMDAS) and how to use parentheses when you want a different order?
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If you’re not sure, work the rest of exercises and then return to these questions. Or, ask your instructor or a classmate for help.

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Exercises Exercises

Exercises 1-4 are available in a separate workbook format.

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1.

In 2011,

Aside

the GDP (gross domestic product) of the United States was approximately $15,596 billion in 2011 and the population of the United States was approximately 0.313 billion that year.

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(a)

Writing the population as 0.313 billion seems strange. A more natural unit would be millions. Rewrite the population in millions of people.

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(b)

Rewrite the population in people, both in normal decimal notation (that means with all the 0s) and in scientific notation.

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(c)

It also seems strange to write the GDP as $15,596 billion. A more natural unit would be trillions, where 1 trillion = 1,000,000,000,000. Rewrite the GDP in trillions of dollars.

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(d)

Rewrite the GDP in dollars, both in normal decimal notation and in scientific notation.

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(e)

Calculate the GDP per capita (meaning per person) by dividing the GDP in dollars by the population in people. Express your answer in $/person.

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(f)

For practice, repeat your calculation using the numbers in scientific notation.

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2.

Edgar recently changed the cleaning bag on his vacuum cleaner.

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He became curious about how many particles of dust were in the bag. Edgar did a little research online and found out that the mass of a dust particle is

\begin{equation*} 0.000~000~000~753\text{ kilograms.} \end{equation*}

(The strange-looking spaces are to help you see that there are 9 zeros in the number.)

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(a)

Write the mass of a dust particle in scientific notation.

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(b)

Recall that

\begin{align*} \amp\textbf{kilo} \amp \amp= \amp \amp 1 \text{ thousand} \amp \amp =\amp 1{,}000\amp\amp \amp=\amp 10^3\amp\\ \amp\textbf{milli} \amp \amp= \amp \amp 1 \text{ in a thousand} \amp \amp =\amp 0\amp.001\amp \amp=\amp 10^{-3}\amp\\ \amp\textbf{micro} \amp \amp= \amp \amp 1 \text{ in a million} \amp \amp =\amp 0\amp.000~001\amp \amp=\amp 10^{-6}\amp\\ \amp\textbf{nano} \amp \amp= \amp \amp 1 \text{ in a billion} \amp \amp =\amp 0\amp.000~000~001\amp \amp=\amp 10^{-9}\amp \end{align*}

Express the mass of a dust particle in each of the given units:

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(i)

grams

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(ii)

milligrams (mg)

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(iii)

micrograms ($\mu$g)

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(iv)

nanograms (ng)

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(c)

Edgar determined that the full vacuum bag weighed 5 pounds. How many dust particles were in the bag? (I am already sneezing.) Use $1 \text{ kilogram} \approx 2.2 \text{ pounds}\text{.}$ Express your answer in scientific notation.

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3.

The list shows the (approximate) mass of the planets in our solar system.

Aside

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Earth	$5.972 \times 10^{24}$ kg
Jupiter	$1.899 \times 10^{27}$ kg
Mars	$6.417 \times 10^{23}$ kg
Mercury	$3.302 \times 10^{23}$ kg
Neptune	$1.024 \times 10^{26}$ kg
Saturn	$5.685 \times 10^{26}$ kg
Uranus	$8.681 \times 10^{25}$ kg
Venus	$4.868 \times 10^{24}$ kg

(a)

Write the mass of Earth and the mass of Mars in standard decimal notation. Which is heavier?

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(b)

List the planets from heaviest (largest mass) to lightest (smallest mass).

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(c)

The mass of astronomical bodies are sometimes measured in Jupiter mass, abbreviated $M_J\text{,}$ where $1 M_J = 1.899 \times 10^{27}$ kg.

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Express Earth’s mass in $M_J\text{.}$

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4.

Souksavanh is setting up a patient’s intravenous (IV) medication.

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She sets the drip at 42 drops/minute. The drip chamber size is 20 drops/mL. Recall

\begin{align*} \amp\textbf{milli} \amp \amp= \amp \amp 1 \text{ in a thousand} \amp \amp =\amp 0\amp.001\amp \amp=\amp 10^{-3}\amp\\ \amp\textbf{micro} \amp \amp= \amp \amp 1 \text{ in a million} \amp \amp =\amp 0\amp.000~001\amp \amp=\amp 10^{-6}\amp \end{align*}

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(a)

At what rate is the IV fluid being delivered to Souk’s patient? Answer in mL/min (millileters per minute).

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(b)

How fast is the drip measured in $\mu$L/sec (microliters per second)?

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(c)

If the drip bag holds 1 liter, how long will it take the drip to run? Express your answer in hours and minutes.

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(d)

The concentration of medication is 1.7 mg/mL (milligrams per milliliter). How much medication is in the 1 liter bag? Convert your answer to grams. Explain what you notice.

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(e)

At what rate is the medication being delivered to Souk’s patient? Answer in g/min (grams per minute).

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5.

(a)

How many files at an average of 42.3 MB each can each gig (1 GB) of computer memory hold?

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(b)

Tara’s coworker Brandon has a much faster Internet connection on his computer at 1,500 kbps. How long would it take Brandon to download 700 MB?

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(c)

At that rate, how much information could Brandon download in 8 hours? Express your answer in kilobytes (KB).

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6.

(Story also appears in 0.8.5)

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(a)

Convert each of these amounts of time into an understandable unit of time: 1 million seconds, 1 billion seconds, 1 trillion seconds.

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(b)

Billy Bob wants to throw a party when he turns 1 billion seconds old. About how many years old will he be?

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(c)

Bonus question: On what date were you or will you be 1 billion seconds old? Don’t forget leap years!

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7.

A proton has mass of about $1.67262 \times 10^{-27}$ kg,

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while an electron has mass of about $9.10938 \times 10^{-31}$ kg.

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(a)

Write out the mass of a proton and that of an electron in normal decimal notation.

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(b)

Which is heavier (has greater mass)?

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(c)

How many times heavier is it? To calculate the answer take the mass of the heavier particle and divide it by the mass of the lighter particle.

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(d)

How many protons would it take to weigh an ounce? Use $1 \text{ ounce} \approx 28.3 \text{ grams}$ and, as always, 1 kg = 1,000 grams. Because $\times$ and $\div$ are at the same level in the order of operations, you should put parentheses around each number in scientific notation before dividing.

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8.

How many servings are in:

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(a)

A 2-liter bottle of a soft drink where the serving size is 250 mL?

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(b)

A 750 mL bottle of wine where a serving size is 5 (fluid) ounces? Use $1 \text{ quart} = 32 \text{ (fluid) ounces}$ and $1 \text{ liter} \approx 1.056 \text{ quarts}\text{.}$

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9.

Rayka weighs 140 pounds.

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She would like to approximate how many cells are in her body. Use the following information: $1 \text{ cell} \approx 1 \times 10^{-15} \text{ g}\text{,}$ $1 \text{ kg} \approx 2.2 \text{ pounds}\text{,}$ and, as always, 1 kg = 1,000 g.

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(a)

How many cells are in Rayka’s body? Write your answer in scientific notation.

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(b)

Rewrite your answer in the most appropriate unit: millions ($10^6$), billions ($10^9$), trillions ($10^{12}$), quadrillions ($10^{15}$), or quintillions ($10^{18}$).

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10.

Body Mass Index (BMI) is one indicator of whether a person is a healthy weight.

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BMI values between 18.5 and 24.9 are considered “normal”. Jared is 6′4″ and weighs 200 pounds. He would like to calculate his BMI from this guide:

\begin{equation*} \text{BMI} = \text{weight in kilograms} \div \text{height in meters} \wedge 2 \end{equation*}

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(a)

Check that Jared is around 1.93 meters tall and weighs around 90.91 kilograms. Use $1 \text{ inch} \approx 2.54 \text{ cm}$ and $1 \text{ kilogram} \approx 2.2 \text{ pounds}$

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(b)

Jared entered the following keystrokes on his calculator:

\begin{equation*} 90.91 \div 1.93 \wedge 2= \end{equation*}

and got the answer

\begin{equation*} \text{Jared's BMI } = 24.4060243\ldots \end{equation*}

Is his BMI considered “normal”?

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(c)

Suppose Jared had rounded off his height to 1.9 meters and his weight to 91 kilograms. Calculate his BMI by entering the following keystrokes your calculator:

\begin{equation*} 91 \div 1.9 \wedge 2= \end{equation*}

What do you get? Round your answer to one decimal place. Is Jared’s BMI considered “normal”?

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(d)

What would you tell Jared?

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Earth	\(5.972 \times 10^{24}\) kg
Jupiter	\(1.899 \times 10^{27}\) kg
Mars	\(6.417 \times 10^{23}\) kg
Mercury	\(3.302 \times 10^{23}\) kg
Neptune	\(1.024 \times 10^{26}\) kg
Saturn	\(5.685 \times 10^{26}\) kg
Uranus	\(8.681 \times 10^{25}\) kg
Venus	\(4.868 \times 10^{24}\) kg