 A modern theory of continued fractions begins with Euler's essay on continued fractions, published in 1744, but written around 1737. In this work, Euler introduced a general continued fraction, where all terms are assumed positive. To evaluate the continued fraction, Euler considered the partial quotients, which he eventually calls by their modern name the convergence. You can think of the convergence as what happens when you leave off a portion of the denominator. Since all the terms are assumed positive, then A will be less than the value of the continued fraction. For the next convergent, since B is smaller than the complete denominator, A plus alpha B will be larger than the value of the continued fraction. For the next convergent, C is smaller than the complete denominator, so B plus beta C's will be larger than the complete denominator, and so the convergent will be smaller than the value of the continued fraction. And so in general, the convergence will alternate between being greater than the value of the continued fraction and less than the value of the continued fraction. Euler used this alternating property of the convergence to prove that continued fractions always converge. While this is true for generalized continued fractions, Euler then focused on continued fractions of the form, where all of our numerators are one. The motivation for this change is that the Euclidean algorithm allows us to convert a rational number a, b, into a continued fraction of this form, where a, b, c, d, and so on are the quotients that result from the Euclidean algorithm. The convergent property means that the continued fraction expansion of a rational number a, b, provides a sequence of successively better rational approximations, which will have smaller denominators. So to convert 235.19's into a continued fraction, we'll apply the Euclidean algorithm. 235 divided by 19 is 12, with remainder 7, then 19 divided by 7 is 2, remainder 5, 7 divided by 5 is 1, remainder 2, 5 divided by 2, 2, remainder 1, 2 divided by 1 is 2, remainder 0, and since we have a remainder of 0, we can stop at this point, and our quotients give what we usually refer to as the denominators of our continued fractions, while our numerators are always one. While the Euclidean algorithm gives us the actual terms of our continued fraction expansion, it's helpful to see why it works. So we can view this as follows. The fraction 235.19's, first we can rewrite that as the mixed number 12 and 7.19's. So what we'll do now is we'll rewrite 7.19's as the reciprocal of 19.7's. But now 19.7's can be rewritten as the mixed number 2 and 5.7's. And again we'll rewrite 5.7's as the reciprocal of 7.5's. And rewrite 7.5's as 1 and 2.5's. Rewrite our fraction as its reciprocal. Rewrite the denominator as a mixed number. And since our last numerator is 1, we've completed the process of converting this into a continued fraction. Our convergence, our first one, is going to be the whole number part. Our next one will include the first fraction. We'll include one more term for our next convergent. And because this is a finite continued fraction, our last convergent is just 235.19's. The important thing here is that these give successively better approximations to 235.19's even though the denominators are smaller.