Part of the confusion may come from poor(?) naming. An irrational number is clearly understood as one that's not representable as a ratio of integers. Non-algebraic works for me, but is a bit of a mouthful.
This is much the same as we call non-composite numbers prime, i.e. numbers missing the property of being composable as a product. Explaining what a prime number is harder than explaining what a composite number is, because it's the negative space. Similarly a transcendental number is the negative space when you remove rational and algebraic numbers from the set of real (or complex) numbers. I don't know of a way to describe such a negative space in a direct/'positive' way.
This is much the same as we call non-composite numbers prime, i.e. numbers missing the property of being composable as a product. Explaining what a prime number is harder than explaining what a composite number is, because it's the negative space. Similarly a transcendental number is the negative space when you remove rational and algebraic numbers from the set of real (or complex) numbers. I don't know of a way to describe such a negative space in a direct/'positive' way.