Algebra

The variable

A quantity that takes different numerical values.

The constant

A quantity that always takes the same value, at least throughout a process. We may think of it as a special case of variables. An example is:

The number scale

An infinite straight line on which there is a point O called the origin (which represents the number 0). A positive direction is chosen, and a constant unit of length separating every two successive integers. Every point in the number scale represents a unique real number, and every real number is represented by a unique point on the number scale.

NB It is customary, although not necessary, to put the number scale horizontally and the positive direction to the right.

Rational numbers

Numbers that can be put in the form of (p/q), where p and q are integers, . They include also all integers (3 for example can be put in the form of 3/1). When they are written as decimals, the decimals will be terminating or recurring nonterminating decimals.

Irrational numbers

Numbers that cannot be put in the form (p/q), where p and q are integers, . They include the roots of some numbers such as  and . They also include some values like  and  (the base of the natural logarithms). Sometimes, they are put in the form of an infinite, but not a recurring decimal.

R

The set of real numbers

The set containing all rational numbers and irrational numbers. Every point in the number scale represents a unique real number, and every real number is represented by a unique point on the number scale.

Interval

The set of all real numbers x lying between two real numbers a and b for example including a or b or both or neither.

An open interval is denoted by ]a,b[. We say that:

.

A closed interval is denoted by [a,b]. We say that:

.

A semi closed interval is denoted by [a,b[ or ]a,b], depending on which point is excluded:

If the interval has no upper (or lower) bound at one side, it is considered open at that side and we write  or instead of a or b, such as:

 or .

Note that .