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. So, really
\begin{align*}
\textbf{megabyte} \amp= \amp
1 \textbf{ million bytes} \amp=\amp
1{,}000{,}000\text{ bytes}\\
\textbf{gigabyte} \amp= \amp
1 \textbf{ billion bytes} \amp=\amp
1{,}000{,}000{,}000\text{ bytes}
\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{align*}
8{,}460\text{ MB} \amp= 8{,}460\text{ million bytes} = 8{,}460{,}000{,}000 \text{ bytes} \\
\amp= 8.46 \text{ billion bytes} = 8.46 \text{ GB }
\end{align*}
So Tara wants to store less than 9 GB of information and she has 72 GB remaining. She has plenty of room. Save away!
Really large numbers, like 8,460,000,000, are awkward to read and awkward to work with. Words like million and billion or metric prefixes (words like mega and giga) 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*}
10^3=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 $}}_{\text{9 times}}\\
\amp=8.46 \ast 10^9
\end{align*}
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*}
8.46\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*}
You can check that
\begin{equation*}
5 \times 10 \wedge 3 = 5000
\end{equation*}
Back to our large number. Enter
\begin{equation*}
8.46 \times 10 \wedge 9=
\end{equation*}
What do you see? Some calculators correctly show \(8{,}460{,}000{,}000\) while other calculators 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?
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.
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*}
Use your calculator to check!
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, we’ll write \(\times\) instead of \(\ast\) to get:
\begin{equation*}
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, we saw that \(8.46 \times 10^ 9 = 8{,}460{,}000{,}000\) and \(1.2 \times10^{-4} = 0.000~12\text{.}\) 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.
