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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
\(2 \le m \lt \frac{n}{2} \text{.}\)
Aside: See.
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*}
