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Subsection 1.4.1 Even and odd integers
We give definitions of certain types of integers so that we have some objects to play with.
Definition 1.4.1.1.
An integer
\(n\) is
even if there exists an integer
\(k\) such that
\(n = 2k\) .
Proof Technique 1.4.1.2. Show \(n\) is even.
Suppose \(n\) is an integer, and we want to prove \(n\) is even.
-
Find
\(k\) such that
\(n = 2k\text{.}\)
-
Show
\(k\) is an integer.
-
Conclude
\(n\) is even, by definition.
Example 1.4.1.3. Twelve is even.
The integer
\(12\) is even because
\(12 = 2 \cdot 6\text{,}\) and
\(6\) is an integer.
Example 1.4.1.4. Zero is even.
The integer
\(0\) is even because
\(0 = 2 \cdot 0\text{,}\) and
\(0\) is an integer.
Definition 1.4.1.5.
An integer
\(n\) is
odd if there exists an integer
\(k\) such that
\(n = 2k + 1\) .
Proof Technique 1.4.1.6. Show \(n\) is odd.
Suppose \(n\) is an integer, and we want to prove \(n\) is odd.
-
Find
\(k\) such that
\(n = 2k + 1\text{.}\)
-
Show
\(k\) is an integer.
-
Conclude
\(n\) is odd, by definition.
Example 1.4.1.7. Twenty three is odd.
The integer
\(23\) is odd because
\(23 = 2 \cdot 11 + 1\text{,}\) and
\(11\) is an integer.
Example 1.4.1.8. Negative twenty seven is odd.
The integer
\(-27\) is odd because
\(-27 = 2 \cdot (-14) + 1\text{,}\) and
\(-14\) is an integer.
In
Theorem 5.1.1.15 we show that every integer is even or odd and that no integer is both. Thus we have:
