Refresher - Real, Rational, and Complex Numbers and Inequalities

A real number, R, may be either rational Links to an external site. or irrational Links to an external site.. If we let R denote the set Links to an external site. of all real numbers. Then:

(1) The set R is a field Links to an external site., meaning that addition Links to an external site. and multiplication Links to an external site. are defined and have the usual properties.

(2) The field R is ordered Links to an external site., meaning that there is a total order Links to an external site. ${\displaystyle \ge }$ such that, for all real numbers x, y and b
     if${\displaystyle x\ge y}$ then ${\displaystyle x+z\ge y+z}$;
     if ${\displaystyle x\ge 0}$ and ${\displaystyle y\ge 0}$ then ${\displaystyle xy\ge 0}$.

For further explanation, see this short video, The Set of Real Numbers.

Properties of real numbers include identity properties, inverse properties, commutative properties, associative properties, and the distributive property. For further explanation, see this short video.

In mathematics, an inequality is a statement about the relative size or order of two objects. Real numbers, R, are ordered by a relation less than (<).  On a number line, one can say that “a is less than b” (a < b) if the graph of a is to the left of the graph of b. 

Basic properties of order:

• Trichotomy: For any real numbers, a and b, exactly one of the following is true:  or ${\displaystyle a>b}$
• Transitive:  For all real numbers, a, b, and c,
  • (a)  if and , then
  • (b)  if ${\displaystyle a>b}$ and ${\displaystyle b>c}$, then ${\displaystyle a>c}$

• Reversal:  For real numbers,a and b,
  • (a)  if ${\displaystyle a>b}$, then
  • if then ${\displaystyle b>a}$
• Addition and Subtraction:  For any real numbers, a, b, and c, 
  • (a) if ${\displaystyle a>b}$, then ${\displaystyle a+c>b+c}$ and ${\displaystyle a–c>b–c}$
  • (b) if , then and
• Multiplication and Division:  For any real numbers, a, b, and c
  • (a)  If ${\displaystyle c>0}$ and ${\displaystyle a>b}$, then ${\displaystyle ac>bc}$ and ${\displaystyle \frac{a}{c}>\frac{b}{c}}$
  • (b)   If ${\displaystyle c>0}$ and , then and
  • (c) If and ${\displaystyle a>b}$, then and
  • (d) If and , then ${\displaystyle ac>bc}$ and ${\displaystyle \frac{a}{c}>\frac{b}{c}}$

A rational number (commonly called a fraction) is a number that can be expressed as a fraction Links to an external site. ${\displaystyle \frac{a}{b}}$, where a and b are integers Links to an external site. and ${\displaystyle b\ne 0}$. Numbers that are not rational are called irrational numbers Links to an external site.. For example, ${\displaystyle \sqrt{2}}$ is irrational.

The set of rational numbers, Q, together with the addition and multiplication operations, form a field.

A complex number is a number Links to an external site. of the form a + bi where a and b are real numbers Links to an external site., and i is the imaginary unit Links to an external site., with the property i ² = -1. The real number a is called the real part Links to an external site. of the complex number, and the real number b is the imaginary part Links to an external site.. When the imaginary part b is 0, the complex number is just the real number a. Complex numbers can be added, subtracted, multiplied, and divided like real numbers, but they have additional properties. Real numbers alone do not provide a solution for every polynomial Links to an external site. algebraic equation with real coefficients, while complex numbers do (the Fundamental Theorem of Algebra Links to an external site.).

Complex numbers can be added, subtracted, and multiplied by formally applying the associative Links to an external site., commutative Links to an external site. and distributive Links to an external site. laws of algebra, together with the equation i ² = -1. Division of complex numbers can also be defined. However, it is not clear which is greater,3+6i or 4+7i.

To see examples of operations on complex numbers, see these videos, Adding and Subtracting Complex Numbers, Multiplying Complex Numbers, and Dividing Complex Numbers below: