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?
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. (Maybe it’s time for a gigamall?)
\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.
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.
(Okay, I have to mention something here. Kilo by itself is pronounced “KEE-loh,” but kilogram is pronounced “KIL-uh gram,” and kilometer is pronounced “ki-LOM-i-ter.” Well, around these parts at least.)
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.
Quick note. 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. 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.
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.
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*}
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.
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*}
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.
Here’s the full list of the order of operations, the priority ranking for arithmetic operations.
Order of Operations
First, calculate anything inside Parentheses ().
Next, calculate Exponents \(\wedge\text{,}\) in order from left to right.
Then, Multiply \(\times\) and Divide \(\div\text{,}\) in order from left to right.
Last, Add \(+\) and Subtract \(-\text{,}\) in order from left to right.
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.
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.)
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*}
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*}
Enough of that. 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?
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.
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.
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*}
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.
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*}
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.
