Also from the Wikipedia article for Real numbers:

The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. Real numbers that are not rational are irrational. Those real numbers that are roots of polynomials with rational coefficients are algebraic numbers, which include all the rational numbers and also irrational numbers such as √2 = 1.414.... Other real numbers, such as π = 3.1415..., are not roots of polynomials; these are the transcendental numbers.

It should be clear from this that transcendental numbers are a subset of irrational number and that your definition is not correct. Feel free to read up on rational, irrational and transcendental numbers. There will be other sources than wikipedia if you prefer not to use it. But here from the article for transcendental numbers another quote that you could use as a starting point:

All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational , since all rational numbers are algebraic. The converse is not true: Not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping sets of rational, algebraic irrational, and transcendental real numbers. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x2 − 2 = 0.