Why is it not divisible by 0?

Reading Time: 3 minutes

The answer is Hadrian’s Knight:

I’m not going to give you the answer you expect because it is actually possible to divide a non-zero real number by 0.

There are two basic approaches you can take to segmentation:

• Application of Inverse Function (Analytic Approach)
• Inverse search for the “multiplication” rule (algebraic approach).

In the first case, there is a singularity at 0. At this point the inverse function simply cannot be used. We can extend it to premises, but nothing helps, a point remains undefined.

A belief that one may sometimes encounter is an extension of continuity or an analytic extension. This applies to certain functions such as x⟼sin(x)/x. But in our case, the left and right limits are completely different.

In the algebraic approach, the inverse of x for the multiplication rule is the unique element y (if it exists) i.e. xy=1 (neutral for the multiplication rule). If y finds a number like 0×y=1, there is cause for concern because you are contradicting the fact that 0 absorbs.

Rules depend on context

In short, you cannot divide a number by zero. Finally… in science everything is allowed as long as it’s fair! Indeed, in mathematics and physics, the rules depend on context. If you don’t state the context, the rule has no meaning.

For example, the sum of the measures of the angles of a triangle does not always add up to 180 degrees. This is only true if we consider a flat space. For a spherical surface, the sum is greater than 180 degrees and for a hyperbolic surface less than 180.

In elementary grades, most of the rules you learn in math and physics can be broken. But it takes more or less creativity and a broader conceptual framework. I gave you the first simple geometric example.

Another simple example is that a negative number has no square root, or an exponential is always strictly positive. This was no longer true in the final year when the general convention of mixed numbers was introduced. Other beautiful examples can be found in my answer to the question “What are the most fascinating science facts?”.

What about “x/0” sacrilege? Unfortunately, to successfully break this rule without setting everything on fire, you have to put yourself in a setting that requires a little more imagination. It’s projective spaces, and more obviously, a real line compressed in time, here’s a little diagram:

This strange point called “infinity” is actually the limit of all real sequences, which are absolutely increasing and infinite. In this set, all numbers have an inverse, and the inverse of 0 is ∞.

You can use the tangent function to move to a region of length 2π from the real line. All that remains is to glue the ends back together to join the infinities.

For those who have studied a bit of mathematics in higher education, addition is not the law of composition, unfortunately not the composition of the body. We also lose order. Truths are simultaneously less strictly and more than eccentric ∞.

You can do the same in campuses. It gives what is called Riemann sphere:

Bjoern_klipp and GKFX Via Wikimedia Commons

Here, ∞ is what the series going to infinity tends to, regardless of direction. Unless you do math or a bit of advanced physics (say requiring topological elements), these gaps are rarely encountered.

But I hope this has shown you once again that scientists are not uncreative psychos. It may not be poetic, because scientists constructively value the logical coherence of their works. There is less creative freedom than art, but you can still divide by zero! (But not for free.)