Let's look at a number... 8...
What number can we multiply to 8 to get 1 (the multiplicative identity) as the answer?
Wow! That's a really weird problem. Can you think of any integers that would work? Nope. So... It must be a fraction! Remember that we want 1 for the answer... and 1 in fraction language with 8's is
So, the multiplicative inverse of 8 is 1/8!
Let's go the other way...
What number can we multiply to 1/8 to get 1 as the answer?
So, the multiplicative inverse of 1/8 is 8!