Sets of Real Numbers

Grouping or classifying is a familiar technique in the natural sciences for dealing with the immense diversity of things in the real world. For instance, in biology plants and animals are divided into various phyla, and then into classes, orders, families, genera, and species. In much the same way, real numbers can be grouped or classified by singling out important features possessed by some numbers but not by others. By using the idea of a set, classification of real numbers can be accomplished with clarity and precision.

A set may be thought of as a collection of objects. Most sets considered in this tutorial are sets of real numbers. Any one of the objects in a set is called an element, or member, of the set. Sets are denoted either by capital letters such as $$A$$, $$B$$ and $$C$$ or by braces $$\{ \cdots \} $$ enclosing symbols for the elements in the set. Thus, if we write $$\{ 1,2,3,4,5\} $$, we mean the set whose elements are the numbers $$1,2,3,4$$ and $$5$$. Two sets are said to be equal if they contain precisely the same elements.

Sets of numbers and relations among such sets can often be visualized by the use of a  number line or coordinate axis. A number line is constructed by fixing a point $$O$$ called the origin and another point $$U$$ called the unit point on a straight line $$L$$. The distance between  $$O$$ and $$U$$ is called the unit distance and may be $$1$$ inch, $$1$$ centimeter, or $$1$$ unit of whatever measure you choose. If the line $$L$$ is horizontal, it is customary to place $$U$$ to the right of $$O$$.

Each point $$P$$ on the line $$L$$ is now assigned a “numerical address” or coordinate $$x$$ representing its assigned distance from the origin, measured in terms of the given unit. Thus, for $$x = \pm d$$, where $$d$$ is the distance between $$O$$ and $$P$$, the plus sign or minus sign is used to indicate whether $$P$$ is to be right or left of $$O$$. Of course, the origin $$O$$ is assigned the coordinate $$0$$(zero), and the unit point  $$U$$ is assigned the coordinate $$1$$. On the resulting number scale, each point $$P$$ has a corresponding numerical coordinate $$x$$ and each real number $$x$$ is the coordinate of a uniquely determined point $$P$$. It is convenient to use an arrowhead on the number line to indicate the direction in which the numerical coordinates are increasing.

A set of numbers can be illustrated on a number line by shading or coloring the points whose coordinates are members of the sets.

For instance:

  • The natural numbers, also called counting numbers or positive integers, are the numbers $$1,2,3,4,5,$$ and so on, obtained by adding $$1$$ over and over again. The set $$\{ 1,2,3,4,5, \cdots \} $$ of all natural numbers is denoted by the symbol $$\mathbb{N}$$.
  • The integers consist of all the natural numbers, the negatives of the natural numbers, and zero. The set of all integers $$\{ \cdots , – 4, – 3, – 2, – 1,0,1,2,3,4, \cdots \} $$ is denoted by the symbol $$\mathbb{Z}$$.
  • The rational numbers are those numbers that can be written in the form $$\frac{a}{b}$$, where $$a$$ and $$b$$ are integers and $$b \ne 0$$. Since $$b$$ may equal $$1$$, every integer is a rational number. Other examples of rational numbers are $$\frac{{13}}{2}$$, $$\frac{3}{4}$$ and $$ – \frac{{22}}{7}$$. The set of all rational numbers is denoted by the symbol $$\mathbb{Q}$$ (which reminds us that rational numbers are quotients of integers). Rational numbers in decimal form either terminate or begin to repeat the same pattern indefinitely.
  •  The irrational numbers are the numbers that are not rational. Their decimal representation is non-terminating and non-repeating. Examples are $$\sqrt 2 = 1.4142135 \cdots $$, $$\sqrt 3 = 1.7320508 \cdots $$, and $$\pi = 3.1415926 \cdots $$.
  • The union or combination of rational and irrational numbers are the real numbers. The positive real numbers correspond to points to the right of the origin, and the negative real numbers correspond to points to the left of the origin. The set of all real numbers is denoted by the symbol $$\mathbb{R}$$.


Rational Numbers and Decimals

By using long division, you can express a rational number as a decimal. For instance, if you divide $$2$$ by $$5$$, you will obtain $$\frac{2}{5} = 0.4$$, a terminating decimal. Similarly, if you divide $$2$$ by $$3$$, you will obtain $$\frac{2}{3} = 0.66666 \ldots $$, a non-terminating, repeating decimal. A repeating decimal, such as $$0.66666 \ldots $$, is often written as $$0.\overline 6 $$, where the over bar indicates the digit or digits that repeat; hence $$\frac{2}{3} = 0.\overline 6 $$.



Express each rational number as a decimal.
(a) $$ – \frac{3}{5}$$
(b) $$\frac{3}{8}$$
(c) $$\frac{{17}}{6}$$
(d) $$\frac{3}{7}$$


(a) $$ – \frac{3}{5} = – 0.6$$
(b) $$\frac{3}{8} = 0.375$$
(c) $$\frac{{17}}{6} = 2.83333 \ldots = 2.8\overline 3 $$
(d) $$\frac{3}{7} = 0.428571428571428571 \ldots = 0.\overline {428571} $$



Express each terminating decimal as a quotient of integers.
(a) $$0.7$$
(b) $$ – 0.63$$
(c) $$1.075$$

(a) $$0.7 = \frac{7}{{10}}$$
(b) $$ – 0.63 = – \frac{{63}}{{100}}$$
(c) $$1.075 = \frac{{1075}}{{1000}} = \frac{{43}}{{40}}$$