Defined and undefined

In mathematics, undefined has several different meanings, depending on the context:

In ancient times, geometers attempted to define every term. For example, Euclid defined a point as "that which has no part". In modern times, mathematicians recognized that attempting to define every word inevitably led to circular definitions, and therefore left some terms, "point" for example, as undefined (see primitive notion).

POINT (an undefined term) In geometry, a point has no dimension (actual size). Even though we represent a point with a dot, the point has no length, width, or thickness. Our dot can be very tiny or very large and it still represents a point. A point is usually named with a capital letter. In the coordinate plane, a point is named by an ordered pair, (x,y).

LINE (an undefined term) In geometry, a line has no thickness but its length extends in one dimension and goes on forever in both directions. Unless otherwise stated a line is drawn as a straight line with two arrowheads indicating that the line extends without end in both directions. A line is named by a single lowercase letter, , or by any two points on the line, .

PLANE (an undefined term) In geometry, a plane has no thickness but extends indefinitely in all directions. Planes are usually represented by a shape that looks like a tabletop or a parallelogram. Even though the diagram of a plane has edges, you must remember that the plane has no boundaries. A plane is named by a single letter (plane m) or by three non-collinear points (plane ABC).

The expression 0/0 is undefined in arithmetic, as explained in division by zero (the expression is used in calculus to represent an indeterminate form).

0⁰ is often left undefined; see Zero to the power of zero for details.

The set of numbers for which a function is defined is called the domain of the function. If a number is not in the domain of a function, the function is said to be "undefined" for that number. Two common examples are ${\displaystyle f(x)={\frac {1}{x}}}$, which is undefined for ${\displaystyle x=0}$, and ${\displaystyle f(x)={\sqrt {x}}}$, which is undefined (in the real number system) for negative ${\displaystyle x}$.

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