The algebraic expression a-b is not equivalent to b-a because?

Let's say a is 1, and b is 2. Is 1-2 the same as 2-1? I didn't think so. This is because subtraction is the same thing as adding a negative number, so by switching the a and b, you are also switching them between negatives and positives. However, -a-b is the same as -b-a.