Condition of Existence of an Algebraic Fraction

For an algebraic fraction $$ \frac{f(x)}{g(x)} $$ to be defined, the denominator must never be zero: $$ C.E. \ g(x) \ne 0 $$ where f(x) and g(x) are algebraic expressions in the variable x.

This requirement ensures that the fraction has a valid mathematical meaning.

In other words, the condition of existence identifies the restrictions on the variables of an algebraic fraction that prevent division by zero, since dividing by zero is not defined in mathematics.

Thus, an algebraic fraction is defined for all values of the variables that do not make the denominator vanish.

This restriction is called the condition of existence, abbreviated as C.E.

A Practical Example

Consider the algebraic fraction

$$ \frac{x}{x-2} $$

If the denominator becomes zero, the fraction is undefined and has no valid numerical value.

Here, the denominator $(x-2)$ is zero when $x=2$.

Therefore, the condition of existence is:

$$ C.E. \ x \ne 2 $$

The fraction is not defined at $x=2$, because at that point the expression cannot be evaluated.

It is defined for every other value of $x$ except 2.

Explanation. At $x=2$, the fraction involves division by zero: $$ \frac{x}{x-2} = \frac{2}{2-2} = \frac{2}{0} $$ Division by zero is undefined because, by the definition of division, there should exist a quotient $q$ such that multiplying it by the divisor (0) yields the dividend (2): $$ \frac{2}{0} = q \ \Leftrightarrow \ q \cdot 0 = 2 $$ But the product of any number $q$ times zero is always zero, since zero is the absorbing element of multiplication: $$ q \cdot 0 = 0 $$ No product $q \cdot 0$ can ever equal 2. Hence, division by zero is impossible.

And so on.