Area of Rectangle equals Length x Width — Proof ?
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This is indeed a very basic question. But trying to answer it can teach some important things, and perhaps help see how mathematics is built.
In the first section, I try to give several motivations why this is an important question. If you are already aware of that you can skip it and jump directly to the section “Looking for a proof”.
Why is this an important question
Something that is widely used in mathematics is the formula:
Area of rectangle = Length x Width
Rectangles form with triangles the most basic shapes, and rectangles can be considered more fundamental than triangles in a certain way. That’s because the area of any triangle can be computed using the above formula.
The ancient mathematicians used right triangles to compute area of circles. To show that the area of a circle is Pi*r², Archimedes used the method of exhaustion, in which a circle is approximated by a polygon.
The area of a polygon of N sides is the sum of N areas of triangles (see figure below).
And to compute the area of a circle, they used to approximate it with a polygon. Hence Pi was approximated using polygons, which are derived from triangles, which are derived from rectangles.
This idea inspired the founders of calculus to derive the area of any shape from that of triangles and rectangles. The notion of integral is represented and interpreted by the areas of rectangles. The notation of Leibniz for integrals that we still use today is nothing but the sum of rectangles that have an infinitely small width.
And by the way we used this idea in computer graphics to represent any 3D shape using primary shapes like triangles:
With that said, it is very important to know whether the formula we are using to compute the area of a rectangle is correct. If it is not, everything that we built upon it will collapse.
Looking for a proof
First, a rectangle of width w and length l, can be divided into w x l unit squares. If we prove that the area of a unit square is equal to 1. Then the area of the rectangle will be w * l * 1 = w * l and we will be done.
I let you search a proof that the area of a unit square is 1… Probably you won’t find it ! You will rather find it as an axiom, which means a mathematical truth that doesn’t need to be demonstrated.
Why don’t we demonstrate axioms in math ? Well we need to remember what a demonstration is in the first place. A demonstration aims to establish a truth (a theorem) from known truths. If you start with 0 truths then you can demonstrate nothing.
But how do we know that the area of a unit square is 1 without proving it. What if it was equal to 2 or 3 or Pi, … Will the mathematics suddenly crash ?
In fact even if this is an axiom, if the area of a unit square was equal to another value, nothing will change! Why ? Because it is up to us to choose the unit square. If the area of a unit square was equal to 2 then we can divide this into 2 squares, and take these instead as unit squares. There is no absoute unit square, it’s us who choose it, so we choose the one that has area equals to 1.
The most important thing is that the area of a unit square inside this definition is something universal for all rectangles, which means that it is the same if you take any 2 rectangles. For example, a property that characterizes circles is that the value circonference/diameter is a constant, meaning that if you take any 2 circles with different diameters and compute this ratio for both of them, you will get the same value. For rectangles its the ratio
area/(length x width) which is constant.
Mathematics is mostly about abstraction and generalization. Early mathematicians invented numbers to abstract the notion of counting for any object. 1 + 1 = 2 is true for any kind of object, we don’t need to write it for each object. When we want to bring the equality to the real world we place the same object in the left and the right hand side. This is different in physics. Remember when you got a zero for a question where you forgot to add the unit?
Going back to axioms. In mathematics, we can take as axioms any truths we want provided that they don’t lead to a contradiction. In his book Sciences Et Méthodes, the french mathematician Henri Poincaré says:
La définition ne sera donc justifiée, au point de vue purement logique, que quand on aura démontré qu’elle n’entraîne pas de contradiction, ni dans les termes, ni avec les vérités antérieurement admises.
In english:
The definition will therefore be justified, from a purely logical point of view, only when it has been demonstrated that it does not lead to any contradiction, neither in terms nor with the truths previously admitted.
So for example instead of taking the area of a unit square as axiom you can take the area of a rectangle as axiom, and compute the area of a unit square from it, this will be a theorem instead of an axiom.
I think this is off-topic, but maybe if you read Henri Poincaré’s quote you might notice a problem. He says we would prove that an axiom is true only when we prove that it doesn’t lead to any contradiction. So, in our case, we need to prove that the area of a unit square is 1 by examining all what it leads to and proving that there is no contradiction with all the possible consequences. Unfortunately for us, It had been proven that “we can’t prove that a system of axioms will not lead to contradictions”. This was 2 decades after the death of Poincaré, so after the quote above, by the logician Kurt Gödel in 1931.
I think there are many things to add, but let this be enough for this article.
TO BE CONTINUED
