Why can't a denominator of a fraction be zero?

Answer 1

To understand this, look at what happens as the denominator approaches zero. Remember that you can always multiply the numerator and denominator by the same amount (which is equivalent to multiplying the entire fraction by 1):

1/1 = 1

1/.1 = (1x10)/(.1x10) = 10/1 = 10

1/.01 = (1x100)/(.01x100) = 100/1 = 100

1/.001 = (1x1000)/(.001x1000) = 1000/1 = 1000

Notice that as the denominator gets smaller, the value of the fraction gets larger. As the denominator goes to zero, the numerator becomes infinitely large. Many people either have no use for, or are uncomfortable with, the concept of infinity, so they say that a fraction with zero in the denominator is undefined.

Now consider that the numerator and denominator are both algebraic functions, rather than numeric values; for example let the numerator be (4 - z2) and the denominator be (2z - 4). When z = 2, the numerator and denominator both evaluate to zero - but in this case the fraction may still be defined - it depends on which function approaches zero faster - calculus gives us the tools to determine that.