Section 2.1 Using Letters Instead of Numbers
Before we begin, a few comments on notation are apropos.
Suppose your teacher starts a lecture with the statement, “Suppose \(a\) and \(b\) are two positive numbers. Then \(a \lt{} a+b\) and \(b \lt{} a+b\text{.}\)” A common complaint is that we’re making the problem harder. This is often accompanied with a request to replace \(a\) and \(b\) with numbers, just to make it easier to understand. For example, if \(a\) and \(b\) are numbers why not just say that \(a\) is \(23\) and \(b\) is \(17\text{?}\) In that case it is clear that \(23 \lt{} 40\ (=23+17)\) and \(17
\lt{} 40\text{.}\) Why do we have to make it more complicated?
This seems like a reasonable point until you think about it a bit more. Suppose we use the numbers \(a=\frac{\sqrt{11}-\pi}{4}\) and \(b=2.93\times 10^{1000}\) instead? Is it obvious to you that
\begin{equation*}
\frac{\sqrt{11}-\pi}{4}\lt{}\left(\frac{\sqrt{11}-\pi}{4}\right)+\left(2.93\times
10^{1000}\right)
\end{equation*}
or
\begin{equation*}
2.93\times
10^{1000}\lt{}\left(\frac{\sqrt{11}-\pi}{4}\right)+\left(2.93\times
10^{1000}\right)?
\end{equation*}
Our point here is that the symbol \(a\) is no different than the symbol \(23\text{.}\) Or the symbol \(\frac{\sqrt{11}-\pi}{4}\text{.}\) These are all valid symbols representing numbers. So we treat them all the same. For example, all of the following are valid statements:
\begin{align*}
a+b &= b+a\\
23+17 &=17+23, \text{ and }\\
\left(\frac{\sqrt{11}-\pi}{4}\right) +\left(2.93\times 10^{1000}\right) &= \left(2.93\times 10^{1000}\right) +\left(\frac{\sqrt{11}-\pi}{4}\right).
\end{align*}
The difference is this: If \(a=23\) and \(b=17\) the first statement includes the second as a specific case. Similarly if \(a=\frac{(\sqrt{11}-\pi)}{4}\) and \(b=2.93\times 10^{1000}\) the first statement includes the third. The first statement is more general. It is true no matter what \(a\) and \(b\) are.
Problem 2.1.1.
(a)
Use the fact‘ that \(\sqrt{7}\lt{}\sqrt{9}\) to show that \(\frac{\sqrt{7}-\pi}{5}\lt{}0\text{.}\) Do not use a calculator.
(b)
Check the validity of the following statements. Are they true?
\begin{equation*}
\frac{\sqrt{7}-\pi}{5}\lt{}\left(\frac{\sqrt{7}-\pi}{5}\right)+2.93\times 10^{1000}
\end{equation*}
\begin{equation*}
2.93\times 10^{1000}\lt{}\left(\frac{\sqrt{7}-\pi}{5}\right)+2.93\times
10^{1000}
\end{equation*}
\begin{equation*}
2.93\times 10^{1000}\lt{}\left(\frac{\sqrt{7}-\pi}{5}\right)+2.93\times
10^{1000}
\end{equation*}
Hint.
Using a calculator on \(2.93\times 10^{1000}\) would be rather pointless. Try it and see.
(c)
Show that it is not always true that if \(a\) and \(b\) are numbers then
\begin{equation}
a\le a+b.\tag{2.1}
\end{equation}
(d)
Find a restriction on the possible values of
\(a\) and
\(b\) that will guarantee that
inequality (2.1) is true.
Keeping things simple is laudable as a goal and we don’t want to suggest that you shouldn’t do it. It can be very useful to look at a special case of a general statement just to get a sense of the problem. For example, a good way to “trick yourself into abstraction” is to think \(23\) and \(17\text{,}\) but write down \(a\) and \(b\text{.}\) As long as you don’t try to do anything that relies on the particular properties of \(23\) and \(17\text{,}\) you should be fine. But you wouldn’t want to say something like \(a+b\) is an even integer (even though \(23+17\) is). The sum \(23+17\) is an even integer because both \(23\) and \(17\) are odd integers, but this is special to these numbers and won’t be true for \(a\) and \(b\) in general.
Problem 2.1.2.
(a)
Suppose \(a\) and \(b\) are both integers. Find restrictions on the possible values of \(a\) and \(b\) that will guarantee that \(a+b\) is an even, positive integer.
(b)
Now suppose \(a\) and \(b\) are both numbers. Find restrictions on the possible values of \(a\) and \(b\) that will guarantee that \(a+b\) is an even, positive integer.
Hint.
This is not the same problem as part (a). Read it carefully.
When looking at \(23\times17\) most people are inclined to compute this to get \(391\text{.}\) If the problem is: Find the value of \(23\times17\text{,}\) then sure, doing the arithmetic is the correct thing to do. But the last time you were asked to do multiplication for its own sake you were probably \(11\) years old. Elementary computations like this will only come up in the context of a more advanced problem. Before doing this calculation, you should ask yourself if it will help you solve the more advanced problem.
Now suppose the problem is: Determine if the number \(23\times17\) is divisible by \(17\text{.}\) Do you see that computing \(23\times17=391\) will only make the problem harder?
Your college professors will never ask you to do an arithmetic calculation for its own sake. Instead arithmetic problems will come up in the context of a more advanced problem and you will need to use your arithmetic skill to uncover patterns. But doing the arithmetic, boiling everything down to a single number, usually hides the patterns!
For example, is it easier to see the pattern in the following list of numbers when they are presented like this:
\begin{equation*}
1, 2, 10, 110, 220, 1100, 12100, 24200, 121000, 1331000, \ldots
\end{equation*}
or like this:
\begin{equation*}
(2^05^011^0), (2^15^011^0),
(2^15^111^0), (2^15^111^1),
(2^25^111^1),
(2^25^211^1), (2^25^211^2), \ldots\text{.}
\end{equation*}
Problem 2.1.3.
Actually, there is something rather liberating about using letters to represent a broad class of numbers. Consider the following questions:
(a)
Is the number \(2^{24}5^{49}11^{1002}\) a perfect square? If so, what is its square root?
(b)
What about \(2^{24}5^{50}11^{1002}\text{?}\) Was your calculator of any use on this? Notice that these numbers were completely factored into their prime factorizations. Did that help? How?
(c)
Now suppose that \(n\) is a positive integer. Is \(2^{2n}5^{3n}11^{4n}\) always a perfect square? If not, for which values of \(n\) will it be a perfect square?
(d)
What can you say, in general, about whether or not an integer greater than one is a perfect square in terms of its prime factorization?
