By H. O. Georgii

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The following expressions are arithmetical. 1918612 − 3 · 1107712 = −2 √ 3+2 2 √ 8−3 7 An arithmetic identity, with a surprising cancellation. The ratio of two quadratic surds having distinct radicands. If indeterminates are present, we speak of an algebraic expression. 6 ab − (ab)−1 √ 3 2 2 a b + ab + 1 An algebraic expression with two indeterminates and higher-index roots. Polynomials and rational functions are algebraic expressions (see Sect. 1). They may also be characterised as rational expressions, since they don’t involve fractional powers of the indeterminates.

6. 7. 8. 9. The set of negative odd integers. The set of natural numbers with three decimal digits. The set of rational numbers which are the ratio of consecutive integers. The set of rational points in the closed unit cube. The complement of the open unit disc in the complex plane. The set of vectors of unit length in three-dimensional euclidean space. The set of circles in the plane, passing through the origin. The set of hyperbolae in the plane, whose asymptotes are the coordinate axes. The set of lines tangent to the unit circle.

X1 ), (x1 , x2 ), (x1 , x2 , x3 ), . ) ({ω}, {{ω}}, {{{ω}}}, . ). 2. 1. f : R → R, x → x + 1 ∞ f (x) dx g(x) 2. 0 d d f (x) dg(x) f (x)g(x) = g(x) + f (x) dx dx dx x d f (y) dy 4. f (x) = dy 3. 0 5. F(x, y)dx 1 6. 1 H (x, y)dy dx 0 0 ∞ (−1)k x 2k (2k)! k=0 ∂ F(x, y, z) ∂ F(x, y, z) ∂ F(x, y, z) + + 8. ∂x ∂y ∂z n ∂L (θ1 , . . , θn ) 9. ∂θk 7. cos(x) = k=1 xn 10. 2 k 1 ∞ x t−1 e−x dx 11. 0 1− 12. π x n 1 x2 n2 . 5 Some Advanced Terms We introduce some advanced words and symbols concerning sets, sequences, and equations.