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Values of Symbols for Which Fractions are Undefined

Here we simply remind you that several properties of fractions of the following special forms continue to apply to fractions in which the numerator and/or denominator are algebraic expressions. We have already discussed and illustrated these properties with reference to numerical fractions. They are

  as long as b is not also equal to zero.
     
  is undefined (or infinite) as long as b is not also equal to zero.
and    
  is undefined (or indeterminate)

The words “undefined” or “indeterminate” here do not mean that people are presently not clever enough to figure out what these expressions must mean. Instead, they indicate that such expressions are numerical nonsense – it is impossible to interpret them in a way that makes numerical sense, or that assigns to them an actual meaningful numerical value.

From this list, we see that essentially, whenever the denominator of a fraction evaluates to zero, we may speak of the value of that fraction as being undefined.

The slight twist with algebraic fractions is that if the denominator is an algebraic expression, then the value of the denominator depends on the values assigned to any symbols that are present. For some values of those symbols, the denominator may evaluate to a nonzero value and so no problem arises. However, there may be some values of the symbols for which the denominator does evaluate to zero, and so the fraction as a whole becomes undefined.

 

Example 1:

For what values of x is the fraction undefined?

solution:

Fractions are undefined when the denominator has a value of zero. Thus, this fraction will be undefined for any value of x that makes

4x – 7 = 0

But this is just a very simple equation. Solving, we get

4x = 7

and so

Checking

4(1.75) – 7 = 7 – 7 = 0.

Thus, the given fraction is undefined only when x = 7 / 4 = 1.75

 

Example 2:

For what values of x is the fraction undefined?

solution:

Again, we need only look at the denominator. The fraction as a whole will be undefined whenever the denominator is equal to zero, that is, whenever

3x 2 – 48 = 0

This is an equation for x which is not too hard to solve:

3x 2 – 48 = 0

gives

3x 2 = 48

and so

So any value of x that can be squared to give 16 will satisfy this equation. But 16 = 4 2 and 16 = (-4) 2 , so there would appear to be two values of x that satisfy this equation and hence make the denominator of the original equation equal to zero:

So, the original fraction is undefined if x = +4 and if x = -4. (You can check by direct substitution that both of these values of x result in the original fraction having the form b / 0, where b 0.)

A final reminder: fractions are undefined in the way we’ve been discussing here only when the denominator evaluates to zero. If the numerator evaluates to zero, but the denominator is nonzero, the fraction itself just has the value of zero and no problem results. Thus, in Example 1 above, if we substitute x = -3 / 5, we get:

This poses no mathematical problem. “Undefinedness” of the sorts referenced in items (ii) and (iii) of the list at the beginning of this note only arises when the denominator evaluates to zero.

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