Chalebgwa, Taboka PrinceMorris, Sidney A.2026-01-302026-01-302025-09-18Taboka Prince Chalebgwa & Sidney A. Morris (2025) Complex Numbers as Powers of Transcendental Numbers, The American Mathematical Monthly, 132:9, 913-917, DOI: 10.1080/00029890.2025.2540754.0002-9890 (print)1930-0972 (online)10.1080/00029890.2025.2540754http://hdl.handle.net/2263/107706It is well-known that if a, b are irrational numbers, then ab need not be an irrational number. Let M be a set of real numbers. In this note it is proved that if M is any of (i) the set of all irrational real numbers, (ii) the set of all transcendental real numbers, (iii) the set of all non-computable real numbers, (iv) the set of all real normal numbers, (v) the set of all real numbers of irrationality exponent equal to 2, (vi) the set of all real Mahler S-numbers, (vii) or indeed any subset of R of full Lebesgue measure, then, for each positive real number s = 1, there exist a, b ∈ M such that s = ab. The analogous result for complex numbers is also proved. These results are proved using measure theory.en© 2025 The Author(s). Published with license by Taylor & Francis Group, LLC. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/),Irrational numbersReal numbersMahler S-numbersLebesgue measureArticle