Unit systems in physics and why we use them

Physics is deeply rooted in equations. Every principle, from Newton's laws to Einstein's theory of relativity, is represented as a mathematical balance between the terms on the left and right sides. This balance involves not just numbers but also the relationships between physical quantities. For an equation to be valid, the way we measure these quantities must be consistent. This requirement is so fundamental that it is often taken for granted. Yet without expressing physical quantities in compatible units, physics would immediately collapse into nonsense. This is why unified systems of measurement are in place.

Fig. 1. Student: I followed the formula. Pythagoras: You followed it where it does not belong.

Consistency in measurement units is essential in all exact sciences. In mathematics, we can't perform operations on fractions until we bring them to a common denominator. Similarly, in geometry, lengths must be converted to the same units, such as meters or feet, before meaningful comparisons can be made. The consequences of ignoring this principle are humorously illustrated in the popular meme shown in Fig. 1. The Pythagorean theorem, which is widely used in physics and engineering, is effectively “broken” when one side of a right triangle is expressed as a real number and the other as an imaginary number. This numerical mismatch leads to an absurd result: the hypotenuse is reduced to zero, effectively collapsing a two-dimensional triangle into a one-dimensional line.

The number i represents the imaginary unit, defined as the square root of negative one: i =√−1. If we remain within a consistent numerical domain, the absurdity immediately disappears. When both legs of the right triangle are expressed as real numbers, 1 and 1, the Pythagorean theorem yields the meaningful result √2, which Euclidean geometry predicts, and physical measurement confirms.

Likewise, if we work consistently within the domain of imaginary numbers, the result is also perfectly coherent. If both legs are expressed in imaginary units, i and i, the Pythagorean relation produces a hypotenuse equal to i√2. Although such a triangle does not correspond to a physical object in ordinary space, the mathematical structure remains internally consistent. The calculation does not break down because the underlying assumptions are respected.

In the meme, the absurd result that a triangle is geometrically equivalent to a line arises from mixing incompatible mathematical frameworks. Just as mathematical operations must remain within compatible domains, physical quantities must share compatible units. In this sense, the meme highlights a fundamental principle: whether we manipulate numbers in mathematics, apply theorems in geometry, or solve physics equations, coherence in units is what keeps these disciplines connected to physical reality.

There exist several unit systems. The most commonly used is the metric SI (International System of Units), which holds official status in almost every country worldwide (Fig. 2). Choosing a particular system is largely a matter of convenience, since all properly defined systems yield correct results. The key requirement is consistency: all quantities must be expressed within the same framework so that comparisons and calculations remain meaningful.

Fig. 2. The seven SI base units that form the foundation of all physical measurements

Consider a simple example. Suppose we want to compare the lengths of two rods. If one length is given in meters (SI) and the other in yards (Imperial system), we can't directly compare the numerical values. We must first convert both measurements to the same unit, whichever we prefer. Only then does the comparison become physically valid. Without this step, the numbers alone are misleading. The same principle applies to every equation in physics. Before plugging numbers into formulas, we must ensure that all quantities use compatible units. Otherwise, even a mathematically correct calculation can produce a physically incorrect result.