# Why are even numbers closed under multiplication?

I'm a little bit confused about why even numbers are closed under multiplication.

Closure under multiplication is $$\lambda u \ \in V$$ where $$V$$ is a vector space. If $$\lambda = 1.5$$, then $$\lambda u$$ is an odd number right? Therefore, it should not be closed under multiplication?

• 1.5 isn't even. – person Oct 12 '20 at 18:34
• Even integers are only closed under multiplication by even integers. And no need to mix it with vector spaces. – Dietrich Burde Oct 12 '20 at 18:35
• The closure under multiplication is meant for a group or a ring, not a vector space. – Ninad Munshi Oct 12 '20 at 18:38
• Does $\lambda$ necessarily have to be even? I thought just the result has to be even. – CSCSCSCSCSCSCSCSCS Oct 12 '20 at 18:43

A closed binary operation merely means that the elements remain in the same set, which is to say the operation is a function of the form $$X \times X \rightarrow X$$. So for example the natural numbers are closed under addition because when you add two naturals numbers together the answer is still a natural number.
In this case the even numbers are closed under multiplication because $$2m \times 2n = 2(2mn)$$ which is even.