Describing Expressions - Essential Dictionary II

Mathematical Writing - Vivaldi Franco 2014

Describing Expressions
Essential Dictionary II

We expand our dictionary with terms describing broad attributes of expressions.

An expression involving numbers, the four arithmetical operations, and raising to an integer or fractional power (extraction of roots), is called an arithmetical expression. The value of an arithmetical expression is a number. A combination of rational numbers and square roots of rational numbers is called a quadratic irrational or a quadratic surd. The following expressions are arithmetical.

$$\begin{aligned} 191861^2-3\cdot 110771^2=-2&\qquad \textit{An arithmetic identity, with a surprising}\\&\qquad \textit{cancellation.}\\ \frac{3+2\sqrt{2}}{8-3\sqrt{7}}&\qquad \textit{The ratio of two quadratic surds having}\\&\qquad {\textit{distinct radicands.}} \end{aligned}$$

If indeterminates are present, we speak of an algebraic expression.

$$\begin{aligned} \frac{\root 6 \of {ab-(ab)^{-1}}}{\root 3 \of {a^2b^2+ab+1}}&\quad {\textit{An algebraic expression with two indeterminates}}\\&\quad {\textit{and higher-index roots.}} \end{aligned}$$

Polynomials and rational functions are algebraic expressions (see Sect. 3.1). They may also be characterised as rational expressions, since they don’t involve fractional powers of the indeterminates.

$$\begin{aligned} \frac{1}{x+1}+\frac{1}{x^2+1}+\cdots +\frac{1}{x^n+1}&\qquad {\textit{The sum of finitely many rational}} \\&\qquad {\textit{functions, with increasing degree.}}\\ \frac{\left( \left( x^2+1\right) ^2+1\right) ^2+1}{(x^2+1)^2+1}&\qquad {\textit{A rational expression, involving}} \\&\qquad {\textit{repeated compositions of a polynomial}}\\&\qquad {\textit{function with itself.}} \end{aligned}$$

The following mathematical Russian doll

$$ \sqrt{x+\sqrt{2x+\sqrt{3x+\cdots +\sqrt{nx}}}}\qquad n\in \mathbb {N}$$

could be described as

An algebraic expression in one indeterminate, featuring a finite number of nested square roots.

The functions sine, cosine, tangent, secant, etc., are called trigonometric functions (or circular functions). A trigonometric expression is an expression containing trigonometric functions.

$$\begin{aligned} 8\cos (z)^4-8\cos (z)^2+1&\qquad {\textit{A quartic trigonometric polynomial.}} \end{aligned}$$

Trigonometric functions belong to the larger class of transcendental functions, which are functions not definable by an algebraic expression (the exponential, the logarithm, etc.). In the expressions (3.16), we have used the terms algebraic, trigonometric and transcendental to describe equations.

The term analytical expression is used in the presence of infinite processes.

$$\begin{aligned} \root 3 \of {1+x}=1+\frac{1}{3}x-\frac{1}{9}x^2+\frac{5}{81}x^3+\cdots&\qquad {{ The\ first\ few\ terms\ of\ the\ series}}\\&\qquad {{ expansion\ of\ an\ algebraic\ function.}} \\ \lim _{n\rightarrow \infty }\,\left( 1+\frac{1}{n}\right) ^n={\text {e}}&\qquad {{ Napier's\ constant\ as the limit of a}}\\&\qquad {{ rational\ sequence.}} \\ 2\prod _{k=1}^\infty \frac{(2k)^2}{(2k)^2-1}=\pi&\qquad {{ An\ infinite\ product\ formula\ for}}\\&\qquad {{ Archimedes'\ constant.}} \end{aligned}$$

An integral expression is an expression involving integrals.

$$\begin{aligned} \ln (x)=\int \limits _1^x\frac{1}{t}\mathrm{{d}}t&\qquad \end{aligned}$$

The term combinatorial is appropriate for expressions involving counting functions, such as the factorial function, or the binomial coefficient.

$$\begin{aligned} \frac{1}{2^{2k}}\frac{(2k)!}{(k!)^2}&\qquad \\ \sum _{k=0}^n{n +k \atopwithdelims ()k}&\qquad \end{aligned}$$

In Chap. 4 we deal with the expressions found in logic: the boolean expressions.

Exercise 3.2

For each expression, provide two levels of description: [ $$\not \varepsilon \,$$ ]

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Exercise 3.3

Same as in Exercise 3.2.

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Exercise 3.4

Same as in Exercise 3.2.

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