Dimensional analysis in physics and the most elegant proof of the Pythagorean theorem.

By: Ali Hamadeh

In physical sciences such as mechanics, electricity, electromagnetic waves, and thermodynamics, Dimensional Analysis is used to verify mathematical equations that link various physical quantities by making sure that the two sides of the equation are "homogeneous" in terms of dimensions.

The concept of dimension can be simplified by replacing it with the ”unit of measurement” used to measure the dimension in which case we replace lengths with meters, time with seconds, masses with kilograms, and so on..

(Replacing the dimension with a unit of measurement is not accurate from a strict point of view, but it allows explaining the idea in a simple way and without going too deep for non-specialists).

Illustrative example:

We write the distance equation for a body moving at a uniform velocity, starting from the origin of coordinates as follows:

x (distance) = v (speed) * t (time)

Now let’s replace the symbols with their units of measurement, we obtain:

m (meters) = meters /sec * sec

By simplifying the unit of measurement of time (sec) between speed (meter / sec) and time (sec) we find that the dimension (unit) of the product of speed by time is homogeneous with the dimension (unit) of distance because both are measured in the same unit which is the meter (see attached picture - top half).

This is a simply explanation of the idea of dimensional analysis.

Let us move now to the most elegant proof of the famous Pythagorean theorem and that is based on dimensional analysis.

The Pythagorean theorem states that the square of the hypotenuse in a right triangle is equal to the sum of the squares of the two right sides.

So we write:

C^2 = A^2 + B^2

Where “C” is the length of the hypotenuse, “A” is the length of the first right leg, and “B” is the length of the second right leg.

This famous theorem has more than 300 mathematical proofs, and here we will present one of them, which in my opinion is the most beautiful or the most elegant (subtle), using dimensional analysis.

To fully define a right triangle, it is sufficient to know the length of the hypotenuse and the value of one of the two angles (other than the right angle which has a value of 90 degrees).

Let's choose one of these angles and call it "x" and try to calculate the area of the right triangle whose hypotenuse is "C" and one of its angles is "x".

Since we are talking about the area, the area equation must contain the length of the hypotenuse raised to the second power "C^2", that is, the square, because the dimension of the area (its unit of measurement) is the square meter when we measure the length of the hypotenuse in meters.

So the area equation will be the product of the square of the hypotenuse by a function and let's call it “f” that depends only on x and which we don’t really need to find the exact mathematical expression.

Hence we can write:

Area of triangle T = C^2 * f(x)

(See attached picture - lower half)

By applying the same logic to triangles T1 and T2, which we obtain by drawing the height from the point of the right angle on the hypotenuse, and noticing that the angle x appears in both triangles, we can write:

Area of Triangle T1 = B^2 * f(x)

Where B is the hypotenuse of right triangle T1

Area of Triangle T2 = A^2 * f(x)

Where A is the hypotenuse of right triangle T2

But the area of the original triangle T, whose hypotenuse C is the sum of the areas of T1 and T2, so we write:

T = T1 + T2

By replacing T1 and T2 expressions we found above, we write:

C^2 * f(x) = B^2 * f(x) + A^2 * f(x)

By canceling f(x) from both sides of the equation, we have:

C^2 *= B^2 + A^2

Which is the exact equation of the Pythagorean theorem in a right triangle.