Blog Post 8

\(1\) \(\underline{\text{Parity}}\)

Parity, in terms of mathematics, describes the classification of an integer as either even or odd. An even number is defined as an integer that is divisible by \(2\) while an odd number is one that is not. A more formal definition states that an even number is an integer \(n\) of the form \(n=2k\) where \(k\) is an integer. On the other hand, an odd number is an integer of the form \(n=2k+1\). In set notation we see:

\[\text{Even} \hspace{0.5mm} = \hspace{0.5mm} {2k : k \in \mathbb{Z}}\]

\[\text{Odd} \hspace{0.5mm} = \hspace{0.5mm} {2k+1 : k \in \mathbb{Z}}\]

In number theory, the idea of parity allows us to solve some mathematical problems simply by making note of odd and even numbers. In the same way, the impossibility of some mathematical constructions can be proven. For example, consider the following question:

\(\hspace{1cm}\) Is it possible to fill in the blanks below with \(+\) and \(-\) signs to create a total of \(20\)?

\(\hspace{1cm}\) \(1 \)_\( 2 \)_\( 3 \)_\( 4 \)_\( 5 \)_\( 6 \)_\( 7 \)_\( 8 \)_\( 9\)

In this case, we find that such a construction is not possible. Here, we have \(5\) odd numbers and therefore the total must be odd. Since \(20\) is even, this proves it cannot be done.

This example makes reference to some properties of the parity of numbers that should be noted. In particular, notice that:

\(\underline{Addition}\) \(\hspace{7.5cm}\) \(\underline{Multiplication:}\)

  • even \(\pm\) even \( =\) even \(\hspace{5cm}\) even \(\cdot\) even \(=\) even

  • even \(\pm\) odd \(=\) odd \(\hspace{5.4cm}\) even \(\cdot\) odd \(=\) even

  • odd \(\pm\) odd \(=\) even \(\hspace{5.5cm}\) odd \(\cdot\) odd \(=\) odd