Arithmetic - Essential Dictionary I

Mathematical Writing - Vivaldi Franco 2014

Arithmetic
Essential Dictionary I

The notation for arithmetical operations is familiar and established. The sum and difference of two numbers $$x$$ and $$y$$ are always written $$x+y$$ and $$x-y$$, respectively. By contrast, their product may be written in several equivalent ways:

$$\begin{aligned} xy\quad \mathrm{} \quad x\cdot y\quad \mathrm{} \quad x\times y, \end{aligned}$$

(2.10)

and so may their quotient:

$$ \frac{x}{y} \quad \,x/y \quad \,x:y. $$

(The notation $$x:y$$ is used mostly in elementary texts.) Do not confuse the product dot ’$$\cdot $$’ with the decimal point$$.$$’, e.g., $$3\cdot 4=12$$ and $$3.4={17}/{5}$$.

The reciprocal of $$x$$, defined for $$x\not =0$$, is also written in several ways:

$$ \frac{1}{x}\quad 1/x \quad x^{-1} $$

while the opposite of $$x$$ is $$-x$$.

The notation for exponentiation is $$x^{\,y}$$, where $$x$$ is the base and $$y$$ the exponent. Defining exponentiation for a general exponent is a delicate matter, as it requires the logarithmic and exponential functions. The case of a positive integer exponent is easier, because exponentiation reduces to repeated multiplication:

$$ x^{n} \,\mathop {=}\limits ^\mathrm{def}\, \underbrace{x\,\cdots \,x}_{n} \qquad n \geqslant 1. $$

The assignment operator $$\mathop {=}\limits ^\mathrm{def}\,$$ [see (2.2)] indicates that this is a definition, giving meaning to the expression on the left. The use of the under-brace is necessary to specify the number of terms in the product, because all terms are identical. Also note the use of the raised ellipsis$$\cdots $$’ to represent repeated multiplication (or repeated applications of any operator), to be compared with the ordinary ellipsis ’$$\ldots $$’, used for sets and sequences (see Sect. 3.1). Thus

$$ \underbrace{x\cdots x}_{4}=x\cdot x\cdot x\cdot x \qquad \underbrace{x,\ldots , x}_{4}=x, x, x, x $$

whereas the notation $$x\dots x$$ is incorrect.

In integer arithmetic, the symbol ’$$|$$’ is used for divisibility.

$$ 3|x \qquad 3\; {\textit{divides}}\; x \qquad x\; {\textit{is a multiple of}}\; 3. $$

EXAMPLE. Turn symbols into words:

$$ \{x\in \mathbb {Z}\,:\, x\geqslant 0,\,2\mid x\}.$$

BAD:

The set of integers that are greater than or equal to zero, and such that 2 divides them. (Robotic.)

GOOD:

The set of non-negative even integers.

A positive divisor of $$n$$, which is not 1 or $$n$$, is called a proper divisor, and a prime is an integer greater than 1 that has no proper divisors. The acronyms gcd and lcm are used for greatest common divisor and least common multiple. (The expression highest common factor (hcf)—a variant of gcd which is popular in schools—is seldom used in higher mathematics.) Two integers are co-prime (or relatively prime) if their greatest common divisor is 1. Some authors use $$(a,b)$$ for $$\mathrm{gcd}(a,b)$$; this is to be avoided, since this notation is already overloaded.

The following concise notation represents certain infinite sets of integers (here $$k$$ and $$m$$ are any integers):

$$\begin{aligned} k+m{\mathbb {Z}}\mathop {=}\limits ^{\nabla }&\, \{x\in {\mathbb {Z}}\,:\, m|(x-k)\}\nonumber \\ =&\, \{\ldots ,k-2m,k-m,k,k+m,k+2m,\ldots \}. \end{aligned}$$

(2.11)

This definition gives meaning to the symbolic expression $$k+m\mathbb {Z}$$ on the left of the assignment operator, which otherwise would be meaningless (you can’t form the sum or product of an integer and a set!). The two expressions on the right represent the same object. While any of the two would suffice, their combination adds clarity (we shall expand this idea in Sect. 6.4).

This notation is economical and effective:

This is a special case of a more general notation for sums and products of sets, to be developed in Sect. 2.3.