Parity
Hin Chi Lee
Parity can be defined as whether the integer being odd or even, and the relationship between them. Here are the following rules all integers always follow:
Even + Even = Even eg 2 + 2 = 4
Even + Odd = Odd eg. 2 + 3 = 5
Odd + Odd = Even eg. 1 + 3 = 4
Even x Even = Even eg. 2 x 2 = 4
Even x Odd = Even eg. 2 x 3 = 6
Odd x Odd = Odd eg. 3x 5 = 15
The concept of parity can be quite useful particularly in disproving an argument that is based on two variables. We take the parity of an odd number as 1, while the parity of an even number as 0. Parity can also be treated as a modulo(2) operation or binary system. When two odd numbers of parity 1 are added together, they have a parity of 2, which cancels to become 0, hence the sum of two odd numbers is an even number.
𝑃 𝑖 ( ) + 𝑃 𝑖 () = 𝑃 𝑖 ( + )
Here are examples which can be solved by parity rules and arguments.
John and Sam are playing are tearing paper after an exam. They start off with 3 pieces of paper, and every time John picks up a piece of paper, he tears that piece of paper into 3 smaller pieces. In a similar fashion, Sam tears the paper he picks up into 5 smaller pieces. They can tear the paper in any order and however many times they like (eg. John can tear it 10 times while Sam doesn’t do it at all). So the question is: is it possible for there to be a total of 100 pieces of paper in any scenario? Have a think. Answer is on the following page.
1
2
5
1
3
3
4
2
John – tears into 3 pieces
Sam – tears into 5 pieces
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