Why Can’t Physical Quantities with Different Dimensions Be Added Together but Can Be Multiplied?

The distinction between adding and multiplying physical quantities with different dimensions is rooted in the fundamental principles of dimensional analysis. This article explores these concepts to provide clarity on when it is appropriate to add or multiply such quantities.

Dimensional Homogeneity and Addition

When adding physical quantities, they must have the same dimensions. This is due to the principle of dimensional homogeneity, which states that only quantities with the same dimensions can be added together. For example, you cannot add 5 meters (length) and 3 seconds (time) because these represent different physical concepts and their addition is meaningless.

Physical Interpretation: Adding quantities with different dimensions does not have a physical interpretation. For instance, if you attempted to add speed (meters per second) to mass (kilograms), there would be no coherent meaning to the result. Imagine trying to add apples and oranges; it simply doesn’t make sense because they are fundamentally different entities.

Dimensional Compatibility and Multiplication

When multiplying physical quantities, the dimensions can be different. The result of such a multiplication is a new quantity that can be expressed in terms of the combined dimensions. For example:

Multiplying speed (meters per second) by time (seconds) gives distance (meters), a meaningful physical quantity. If you multiply force (newtons) by distance (meters), you get work (joules), which makes sense in physical terms.

Scalar and Vector Products: In multiplication, you can have scalars and vectors interact in ways that produce meaningful results. For instance, calculating the work done by a force along a distance is a physical operation that gives a meaningful result.

Summary and Real-World Application

Addition: Requires quantities to have the same dimensions. If they don’t, the result is meaningless.

Multiplication: Allows for different dimensions to be combined, resulting in a new quantity that is meaningful and can be expressed in terms of dimensions.

This fundamental principle ensures that physical equations maintain coherence and reflect real-world relationships accurately.

Real-World Example: Apples and Nails

Let's consider a practical example to further illustrate the concepts. Take two quantities of dissimilar objects—15 apples and 10 nails. You can add these together as numbers of objects, yielding a total of 25 objects. However, it is meaningless to try to add the dimensions (apples nails) because apples and nails are different types of objects:

Addition: 15 apples 10 nails 25 objects. This is a numerical operation without any physical meaning. Subtraction: 15 apples - 10 nails. This would just leave you with 5 objects, which can be a mix of apples and nails depending on the exact subtraction process. Multiplication: 15 apples * 10 nails 150 objects. While you can perform this multiplication as a counting process, it has no meaningful physical interpretation. Division: 15 apples / 10 nails. The result (1.5) has no coherent physical units, making the division operation meaningless.

In contrast, consider the quantities of distance and time:

Distance and time can be combined in a way that yields a meaningful result. For example, dividing distance by time gives speed (meters per second), which is a meaningful and observable quantity.

This example demonstrates that addition and subtraction of different dimensions are meaningless, while multiplication and division can produce meaningful results in the real physical world.