Units are fundamental concepts in mathematics and physics that represent measurable physical quantities. Each unit has a specific meaning - meters measure length, kilograms measure mass, and seconds measure time. These are distinct entities that cannot be arbitrarily combined. When we try to add different units together, we encounter a fundamental problem that we'll explore mathematically.
Dimensional analysis provides the mathematical framework for understanding unit operations. Every physical quantity has dimensions: length has dimension L, mass has dimension M, and time has dimension T. This system acts as a consistency check for mathematical operations. Addition and subtraction are only valid when the dimensions are identical. For example, 5 meters plus 3 meters equals 8 meters because both have length dimension L. However, adding 5 meters to 3 seconds is undefined because we're mixing length and time dimensions.
The fundamental rule for addition states that addition and subtraction operations require identical dimensions. This means we can only add quantities that have the same dimensional structure. When we add 5 meters plus 3 meters, we get 8 meters because both quantities have the same length dimension. However, attempting to add 5 meters plus 3 seconds is mathematically undefined because we're trying to combine different dimensional quantities - length and time - which have no meaningful sum.
We can prove that different units cannot be added using proof by contradiction. First, we assume the opposite of what we want to prove - that we can add different units, like 5 meters plus 3 seconds equals some quantity X. But what dimension would X have? If X has length dimension, then 3 seconds would need to have length dimension, which contradicts its definition. If X has time dimension, then 5 meters would need time dimension, again a contradiction. Since our assumption leads to logical contradictions, it must be false. Therefore, different units cannot be added.
We have successfully proven that different units cannot be added. Our theorem states that addition A plus B is defined if and only if the dimensions of A and B are identical. We established this through dimensional analysis, formal addition rules, proof by contradiction, and physical interpretation. This fundamental principle ensures consistency in all mathematical and scientific work, preventing meaningless calculations and maintaining the logical structure that underlies mathematics, physics, and engineering. The proof is complete and the theorem is established.