What is x/y/0 ?

1 Answer
Apr 8, 2017

Answer:

#x/(y/0)# and #(x/y)/0# are both undefined, except under special circumstances.

Explanation:

When dealing with arithmetic of ordinary numbers, division by #0# is always undefined, so any resulting expression is undefined.

So both #x/(y/0)# and #(x/y)/0# are undefined.

There are at least two non-ordinary contexts in which it may be defined.

They are called the "real projective line" #RR_oo# and the Riemann sphere #CC_oo#.

Both of these extensions of ordinary numbers add a single point "at infinity" #oo# with some arithmetic rules, but not all arithmetic results in determinate values.

For example:

For any #x in RR_oo#, we have:

#x+oo = oo+x = oo#

For any non-zero #x in RR#, we have:

#x/oo = 0#

#x/0 = oo#

The following expressions are all indeterminate:

#0/0#

#oo/oo#

#oo-oo#

#0*oo#

However, we do find that if #x, y in RR#, with #y != 0# then in the "arithmetic" of #RR_oo# we have:

#x/(y/0) = x/oo = 0#