What is 5 divided by 0?

1 Answer
George C.
Jul 30, 2016

Undefined.

Explanation:

Division by #0# is (almost always) undefined.

Suppose we attempted to assign a value to #5/0#, say #a=5/0#

Then multiplying both sides by #0# we get:

#0 = 5#

Since this is false, there is no such value #a#.

#color(white)()#
Footnote

Why "almost always"? Are there any exceptions?

Suppose we try to add an object to our number system called #oo# which satisfies #1/0 = oo# and #1/oo = 0#. What can we do and not do with this object?

We can use it to provide the value of a function where it would otherwise have a vertical asymptote.

We can perform limited arithmetic on it:

If #a# is any ordinary Real number, then #a + oo = oo + a = oo#

If #a# is any ordinary non-zero Real number, then #a*oo = oo*a = oo#. So for example #5/0 = 5*1/0 = 5*oo = oo#

But expressions like #0*oo# have indeterminate value.

The expression #0/0# is still indeterminate too.

This object #oo# does not fit in nicely with arithmetic as a whole, and should probably not be considered a number.

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