Why is energy lost a negative number?
It's not mandatory to have certain values default to a specific integer, though it is more convenient to perceive those values that way. You may invert such values, however, this results in every value given, to be inverted in order to get the same answer.
It doesn't have to be a negative number.
The way physics and mathematics works (in terms of tangible relationships), is that whatever is there/what is going to be there, we assume it as a positive integer. That means what isn't there or what is lost, becomes negative.
Now this is all in the perspective of one individual, it just so happens it makes the most sense for other students.
However, it's not absolutely necessary to have things such as lost energy be a negative. You can make it a positive value and get the same answer, but that means you'll have to invert every single value given. You would also probably have to let the assessor know you've done such things.
Hope this helps :)