Editing Precalculus (section)

==Concept==
For students to succeed at finding the [[derivative]]s and [[antiderivatives]] with [[calculus]], they will need facility with [[algebraic expression]]s, particularly in modification and transformation of such expressions. [[Leonhard Euler]] wrote the first precalculus book in 1748 called ''[[Introductio in analysin infinitorum]]'' ([[Latin]]: Introduction to the Analysis of the Infinite), which "was meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of differential and integral calculus."<ref>{{cite book |last=Bos |first=H. J. M. |author-link=H. J. M. Bos |date=1980 |chapter=Chapter 2: Newton, Leibniz and the Leibnizian tradition chapter 2 |page=76 |title=From the Calculus to Set Theory, 1630 – 1910: An Introductory History |editor-first=Ivor |editor-last=Grattan-Guinness |editor-link=Ivor Grattan-Guinness |publisher=[[Duckworth Overlook]] |isbn=0-7156-1295-6}}</ref> He began with the fundamental concepts of [[variable (mathematics)|variable]]s and [[function (mathematics)|function]]s. His innovation is noted for its use of [[exponentiation]] to introduce the [[transcendental function]]s. The general logarithm, to an arbitrary positive base, Euler presents as the inverse of an [[exponential function]].

Then the [[natural logarithm]] is obtained by taking as base "the number for which the hyperbolic logarithm is one", sometimes called [[Euler's number]], and written <math>e</math>. This appropriation of the significant number from [[Grégoire de Saint-Vincent]]’s calculus suffices to establish the natural logarithm. This part of precalculus prepares the student for integration of the monomial <math>x^p</math> in the instance of <math>p = -1</math>.

Today's precalculus text computes <math>e</math> as the limit <math>e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^{n}</math>. An exposition on [[compound interest]] in financial mathematics may motivate this limit. Another difference in the modern text is avoidance of [[complex number]]s, except as they may arise as roots of a [[quadratic equation]] with a negative [[discriminant]], or in [[Euler's formula]] as application of [[trigonometry]]. Euler used not only complex numbers but also [[infinite series]] in his precalculus. Today's course may cover arithmetic and geometric sequences and series, but not the application by Saint-Vincent to gain his hyperbolic logarithm, which Euler used to finesse his precalculus.