If Simon Stevin already pioneered the unending decimal representation for every number (rational, surd, etc.) at the end of the 16th century, why do Cantor and Dedekind (who certainly gave a more detailed account) routinely get credit for the real numbers?
Stevin did some detailed work (rather than vague general ideas) with unending decimals, including a proof of the intermediate value theorem for polynomials. Newton was in fact inspired by infinite decimals to introduce his general theory of power series.
An interesting point was raised in an answer by Peter Diehr. The so-called Archimedean property (which is one of the defining characteristics of the real number field; though of course it does not suffice to characterize them as the rationals also satisfy it) was considered by authors like Euclid (Elements V.4) considerably earlier. However as far as giving an actual construction (rather than axiomatic definition) Stevin seems to have been the first.
Note 1. To clarify, Stevin developed specific notation for decimals (more complicated than the one we use today) and did actual technical work with them rather than merely envisioning their possibility, unlike some of his predecessors.
Note 2. One useful source for this is Malet, Antoni. Renaissance notions of number and magnitude. Historia Math. 33 (2006), no. 1, 63–81.
Note 3. As Malet notes, "Stevin does not justify his definition" which identifies number and "quantity of anything" because to him the identification is obvious, and the implementation of number is his unending decimals. This was an appropriate move indeed since we know today that the Cantor-Dedekind postulate identifying the number line and the line in physical space is untenable based on what modern physics teaches us; similar remarks apply to magnitude/quantity. Stevin of course was not aware of "transcendental" numbers but no such knowledge is required in order to define the real numbers by means of unending decimals; namely this could have been done even if Liouville did not prove the existence of transcendental numbers.
Note 4. I should clarify that Stevin dealt with unending decimals in his book l"Arithmetique rather than the more practically-oriented De Thiende meant to teach students to work with decimals (of course, finite ones).
Note 5. As far as using the term real to describe the numbers Stevin was concerned with, it should be clarified that the first one to describe the common numbers as real may have been Descartes and at any rate this usage is later than Stevin. On the other hand, if we talk about representing common numbers (including both rational and not so), Stevin not only speculated about the possibility of a representation scheme using decimals, but (unlike some of his predecessors) developed a specific notation (though different from what we use today) and moreover did work with this notation.
Note 6. Cantor thought that Cauchy Completeness (CC) was sufficient to characterize the real numbers axiomatically. Today we know this is not the case, as one also needs the Archimedean property. I found out recently that Dedekind was convinced he had a proof of the existence of an infinite set; see here. Do these misconceptions by Cantor and Dedekind indicate a shortcoming of the constructions of the real numbers they proposed? Hardly so. Stevin's approach to representing all common numbers by unending decimals similarly could not be held at fault because Stevin was not aware of certain future developments.