Is Pi over Pi rational?

Question

A student asked this question in class today, and I wasn't sure of the answer. On the one hand, since Pi is irrational itself, Pi/Pi doesn't fit the definition of a rational number (namely a number of the form a/b where a,b are both integers, b not = to zero). However, Pi/Pi is equivalent to 1, which is certainly rational. Is it most accurate to say that Pi/Pi is irrational (by definition), but that it is equivalent to a rational number? That seems problematic, since it implies a number can be both rational and irrational at the same time.

Answer 1

it is rational given that it could be written as a/b = 1/1.

Answer 2

A better definition to use is that rational number is a number which when multiplied by some non-zero integer gives an integer result. This definition does not require the uneccessary step of selcting which denominator to use in your rational representation.

Since $$ \frac\pi\pi \cdot 1 = 1$$ it meets that definition.

Answer 3

Note that when you write $\frac ab$ you are not actually writing a number, but rather, a certain class of equivalence of pairs by means of some chosen representative.

Over the reals the equivalence is $$ (a,b)\sim(c,d)\Leftrightarrow ad=bc $$ and $\frac ab$ denotes the class of $(a,b)$ (mind that the second element of the pair must be not $0$).

You say that the class $\frac ab$ is rational (i.e. $\in\Bbb Q$) if there exists a pair $\in\Bbb Z\times\Bbb Z$ in that class.

Answer 4

The assertion

Pi/Pi doesn't fit the definition of a rational number

is not true, because

$$\frac{\pi}{\pi}=\frac{1·\pi}{1·\pi}=\frac{1·\require{cancel}\cancel{\pi}}{1·\require{cancel}\cancel{\pi}}=\frac{1}{1}$$

Answer 5

Since $x/x = 1$ for all non-zero $x$, whether rational or irrational, $\pi/\pi$ is likewise equal to $1$.