Elements of Algebra

Apropos 1+2+3+4+5+...=<img src="http://latex.codecogs.com/gif.latex?-\frac{1}{12}" title="-\frac{1}{12}" />: Mapping Infinity in Light of the Number Circle (or Cycle), in L. Euler’s Footsteps and with the Aid of Two Dimensional Infinite Series, and Replacing Negative Infinity and Positive Infinity with Just Infinity

“…But he was mightily impressed by the potential of infinite series and in a rare expression of genuine admiration for a mathematical phenomenon states the following [7]:…”

“…There is in fact more than one way of converting a rational number into an infi- tions" with his usual unsurpassed lucidity in his Elements of Algebra [8].…”

“…Euler does not make a connection between "infinite decimal fractions" and what he describes elsewhere in his Elements of Algebra as "the resolutions of fractions into infinite series" [9]. I will make the connection explicitly further below.…”

“…Euler presents a lucid and delightfully concise description of this phenomenon in the chapter entitled "the resolution of fractions into infinite series" in his Elements of Algebra, which has been described as the second most popular mathematics book of all time, after Euclid's Elements [10]. It has much inspired the following account.…”

“…a a a a a =+ + + + + −  The next step is to evaluate the four infinite sums if a has the values listed in fractions (5), (6), (7), and (8). These values of a are as follows.…”