Ask A Scientist Mathematics Archive

Author:      n scherer
If you divided Infinity by Infinity would you not get 1? If not, could you
explain why?

Response #:  1 of 2
Author:      tee
In the real number system, "INF(INITY)" is not considered to be one of the
set of real numbers and so none of the operations are defined for INF.  Even
in systems where some kind of INF is included (complex numbers in some
contexts) we still see avoidance of defining INF/INF because quotients
supposedly are related to products in a unique way.  For example, define 
a/b = c  if c is the unique number solving the equation a = bx.  If we
define a*INF = INF for all non-zero a (as is reasonably done in some cases)
then INF/INF is not uniquely definable.

Response #:  2 of 2
Author:      jlu
In addition to the previous remarks, INF/INF does seem to occur as the
result of taking limits.  For example, let us divide 1 + x^2 by 1 + x^4 and
ask for the ratio in the limit that x-->INF (x becomes infinite).  Although
the question seems to involve the ratio, INF/INF, the answers is uniquely
zero.  You might try working out some other examples that give different
numbers for the answer.  Try, for example, the ratio of 1 + 3x^2 divided by
1 + x^2 in the limit that x becomes infinite.