Mathematical writing: An undergraduate course by Franco Vivaldi

By Franco Vivaldi

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The exponential and the logarithm are transcendental functions. An equation defined by algebraic expressions is called an algebraic equation. Likewise, we speak of trigonometric and transcendental equations. xn − x − 1 = 0 cos(x) = sin(x) log(1 + x) = −x An algebraic equation. A trigonometric equation. A transcendental equation. The term analytical expression is appropriate in the presence of infinite processes: √ 1 1 5 1 + x = 1 + x − x2 + x3 + · · · 3 9 81 1 n =e lim 1 + n→∞ n ∞ (2k)2 2∏ =π 2 k=1 (2k) − 1 The first few terms of the series expansion of an algebraic function Napier’s constant as the limit of a sequence of rational numbers An infinite product formula for Archimedes’ constant.

The set of all rational numbers which are not the ratio of two consecutive integers. 4. The set of vectors of unit length in three-dimensional euclidean space. 5. The set of circles in the plane, passing through the origin. 6. The set of hyperbolae in the cartesian plane, whose asymptotes are the cartesian coordinate axes. 7. The set of lines tangent to the unit circle. 4. The following expressions define sets. Turn symbols into words. [ ε ] 1. {x ∈ Q : 0 < x < 1} 2. {1/2n : n ∈ N} √ 3. {x + y −1 : x, y ∈ Z} 4.

24) ..  .   fn (x) = gn (x) where all functions have the same domain and co-domain, is called a system of n simultaneous equations in m unknowns. 2, on page 58). Equations are very general objects, and need not be associated to ‘numbers’. 1). We consider the following set equation x ∩ {a} = x ∩ {b} a = b. 25) With the notation introduced above, we have X = Y = P(A) f (x) = x ∩ {a} g(x) = x ∩ {b}. The solution set of the equation is readily seen to be the collection of the subsets x of A which do not contain a or b.

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