Beginning with the essentials from the theory of simple continued fractions, we review some early results in Diophantine approximation. We proceed to prove that Liouville's number L is transcendental; in addition we discuss the transcendence measure of e, known as Davis' theorem, which also gives us e is a transcendental number by Roth's theorem. Having reviewed these results we generalize a result of Landau, giving rise to what we will call Landau sums. These sums generate a countable spectrum of irrational numbers that are computable in quadratic time due to their representation as theta functions; it is still open as to whether these numbers are transcendental. Altering the Landau sums we also get another class of sums that we call lateral Landau sums. These sums also give a spectrum of numbers that we speculate are irrational. These results sit nicely within the intersection of classical analysis and classical number theory. We say a few words about this relationship and end with some conjectures.