Is zero irrational?

A real number $x$ is rational if and only if it can be written as a fraction $a/b$ with $a$ and $b$ integers and $b\neq 0$. In particular, integers are rational, and $0$ is an integer, so it is rational.

One perspective on this is to regard $0$ as a symbol that actually refers to several different mathematical objects:

• $0$-the-natural-number, usually the first natural number defined. Let's write $0_n$ for this specifically.
• $0$-the-integer; an integer is a natural number with a sign, and so we can write $0_i = +0_n$. In fact it's also $-0_n$, but that doesn't really matter.
• $0$-the-rational-number: a rational number is an integer divided by another, so $0_r = 0_i/1_i$

From this point of view, there is nothing circular about $0 = 0/1$, because we're actually just using the same symbol to refer to two very similar objects, one an integer and one a rational number.

That is a good observation, the question is not really about 0 being rational or not but that wether 0 is the only rational with infinitely many representation that can not be simplified.

Use the alternative definition of rationals :Rational can be represented by finite or repeating decimal representation.

Using that definition there is no ambiguity trying to find two numbers that ratio will be 0. But according to that definition there are infinite numbers that their ratio is 0, where we only needed 1. So it is a rational with the property that can be expressed with infinity many different ratios (all other rational have only one ratio in its simplest terms ).