Question

If k is an integer, what is
k + 1
/2
!
? Why?

Answer 1

If $k\in\mathbb{Z}$ than $k+1\in\mathbb{Z}$ however any $\mathbb{Z}$ divided by 2 is in $\mathbb{Q}$.

But it is not so simple.

If $k$ is odd then $k+1$ is even but when even number is divided by 2, you get an odd number which is in $\mathbb{Z}$.

However if $k$ is even then $k+1$ is odd and when we divide that by 2, you get a number like for example $\frac{5}{2}$ which is not in $\mathbb{Z}$ but rather inside $\mathbb{Q}$. There are fortunately no irrational numbers.

There is also a problem with zero, zero is neither odd nor even but still an integer. If your $k$ happens to be -1, $k+1=0$, and zero divided by 2 is still just 0 which is in $\mathbb{Z}$.

So to sum up,

If $k$ is odd and not -1, then $\frac{k+1}{2}\in\mathbb{Z}$.

Or to put it in math $\forall k\in\mathrm{odd}-\{-1\}\cup\{0\}\implies\frac{k+1}{2}\in\mathbb{Z}$.

If $k$ is even or -1, then $\frac{k+1}{2}\in\mathbb{Q}$.

Or to put it in math $\forall k\in\mathrm{even}\cup \{-1\}\implies\frac{k+1}{2}\in\mathbb{Q}$.

Notes:

$\mathbb{Z}$ is a set of integers.

$\mathbb{Q}$ is a set of rational numbers.

$\mathbb{Z}\subset\mathbb{Q}$ means integers are a subset of rational numbers, that is, every integer is a also a fraction, however not every fraction is an integer.

Hope this helps :)