Even Numbers: Closed under addition Odd Numbers: Not closed under addition
Practice makes perfect
The definition of an even number is any number that can be written as 2n, where n is an arbitrary number. An odd number are numbers that are between two even numbers. They can therefore be written as 2n+1.
Even: 2n Odd: 2n+1
Now we can start answering the two-part question.
Are Even Numbers Closed Under Addition?
Let's first define two arbitrary even numbers.
2k and 2m where k≠m
If the sum of these numbers gives an even number, we know that even numbers are closed under addition.
Adding two even numbers gives a sum that can be written as a product where one factor is 2. This fits the definition of an even number. Therefore, even numbers are closed under addition.
Are Odd Numbers Closed Under Addition?
Let's first define two arbitrary odd numbers.
2k+1 and 2m+1 where k≠m
To determine if odd numbers are closed under addition, we will add them and simplify. If they are, the sum will produce another odd number.
Adding two odd numbers gives a sum that can be written as a product where one factor is 2. Therefore, the sum of two odd numbers is not closed under addition.
