You Can't Do That
Grow up! That's the advice of the grownup to the misbehaving youngster, while in all actuality, the desire to return to those carefree days haunts the thoughts and dreams of most over the age of legal driving.
Pertaining to Algebra, we caution young mathematicians when numbers are lined up in columns that they can't subtract 7 in the ones column from the 5 above it unless they borrow from (or rename) the tens column. Later on, the student finds that yes, five minus seven can be done, yielding negative two, a designation often found in my checkbook balance. Then a few grades later, we present square roots, and say you can't take the square root of a negative number. Better texts comment that it will be explained further at a later time, when the concept of imaginary numbers is presented. Finally, the can't concept is applied to dividing by zero. Once again, rather than admit defeat, the feat is called 'undefined.' I would propose a new term. At the moment one calls an item 'undefined' hasn't it become defined? Likewise, when you call an object in the sky a 'UFO' or Unidentified Flying Object, haven't you identified it? But I'm talking about math here, today. The inverse proportional nature of a fraction determines that with a constant numerator, as the denominator becomes smaller, the value of the fraction becomes larger. Ultimately, the smallest denominator possible would be zero, as even negative numbers would have an absolute value greater than zero, and could be factored with negative one into something of worth. Zero as a denominator then would afford a fraction of the largest value, which most students readily identify as 'infinity.' Infinity is an abstract concept, not a value. Hey, that's part of a definition, isn't it? Any attempt to apply functions to a concept result in the return of the concept, for example infinity plus 1 is still infinity, infinity raised to the power of infinity is still infinity and so forth. Even infinity divided by infinity, which should certainly be one, or infinity minus infinity = zero, introduces some uncertainty, to a point of making them for all practical purposes, meaningless. So, when the question comes up on your weekly pop quiz, put down that division by zero is "undefined" or you'll lose points. But smile quietly to yourself, and remember in your heart of hearts, that it's really just meaningless.