Section 6.1 Definitions
Definitions are used in mathematics to label objects that have special properties, and to group all such objects together. Be careful with definitions: as stated by mathematicians, they often contain implicit conditions.
Example 6.1.1.
A number is called prime if its only divisors are \(1\) and itself.
This definition has some hidden parts: a more complete definition would be as follows.
A number is called prime if
it is an integer,
it is strictly greater than \(1\text{,}\) and
there does not exist any other number greater than \(1\) which divides it.
You should view a definition as a technical test or collection of technical tests that an object must pass before it can be given a specific label.
Worked Example 6.1.2.
Demonstrate that, according to the technical definition of prime, \(17\) is prime but \(21\) is not.
Solution.
Let us test \(17\text{.}\)
Yes, \(17\) is an integer.
Yes, \(17 \gt 1\text{.}\)
None of the numbers in the following list is an integer:
\begin{equation*}
\frac{17}{2}, \frac{17}{3}, \frac{17}{4},
\dotsc,
\frac{17}{16}, \frac{17}{18}, \frac{17}{19},
\dotsc\text{.}
\end{equation*}
So \(17\) is prime since it passes the technical tests that define the concept of prime.
Now let us test \(21\text{.}\)
Yes, \(21\) is an integer.
Yes, \(21 \gt 1\text{.}\)
However, clearly \(21/3 = 7\) is an integer, so \(3\) divides \(21\text{.}\)
So \(21\) is not prime, since it fails at least one of the technical tests that define the concept of prime.
Often, the first thing we do in mathematics is to look for ways to make testing our definition easier.
Proposition 6.1.3.
Suppose \(n\) is an integer with \(n \ge 2\text{.}\) Then \(n\) is prime if and only if \(n/m\) is not an integer for every integer \(m\) with \(2 \le m \lt \frac{n}{2}\text{.}\)
Worked Example 6.1.4.
Demonstrate that \(17\) is prime.
Solution. (Sketch)
By the proposition, to check that \(17\) is prime we now only need to note that none of the numbers in the following shorter list is an integer:
\begin{equation*}
\frac{17}{2}, \frac{17}{3}, \frac{17}{4},
\dotsc
\frac{17}{8}\text{.}
\end{equation*}
