## Table of contents

## The Power of the Math Classroom

On a Wednesday morning at 9:00, I go into my 6th grade classroom to work on powers. In this session, specifically, we will discover some of the patterns that appear when we calculate powers that have the same base. For example, we will notice that, in the units of the powers of 2, this pattern is repeated: 2, 4, 8, 6.

The class follows its usual progression, there are moments of effort and frustration, and moments of enjoyment and discovery. Until the bell rings and Ariane shares what she’s been writing in her math book with me… But to really understand it, we need a little more context.

In the previous class, we used a similar pattern to discover the answer of 20: As the exponent decreases by one unit each time, the answer is divided by 2, the answer of 20 has to be the answer of 21 divided by 2, that is, 1.

And in fact, thanks to the discovery of this pattern, we can generalize and say that any natural number to the power of 0 is 1. But then, since we had also discovered that 0 raised to any natural number is 0, Miquel spoke up to ask the big question:

**“So, what is 0 to the power of 0?**”

We put the question to the whole class and two opinions emerged. There were those students who thought 0 to the power of 0 had to be 0, and those who thought 0 to the power of 0 had to be 1. Who was right?

At that moment we discovered that, contrary to what many people think, mathematics is not a closed, perfect discipline. The mathematical community has been unable to answer the answer of 0 to the power of 0. In fact, when you put it to a calculator, you get the message, ‘math error’. Math error! The operation 0 to the power of 0 cannot be solved, because it is not well defined. This left many students with a bittersweet feeling: there are math questions that we don’t have answers for! But this is precisely what makes them extremely interesting.

Meanwhile, Ariane had written the following in her notebook:

*Too bad, a stain has covered up the answer.*

* *

*But that is where the world of mathematics comes into play, a new world that was discovered long ago. But not everyone likes it, because there are those who believe that there are only calculations in this world, calculations, or boring things. But do you really think that this is the case? I don’t know if others like it or not (I don’t really care either), because I love math. Mostly I want to know one thing and prove (even if it’s just to myself) that, if you set your mind to it, you can achieve anything. So I set myself the challenge of 0.*

## Study on Zero

**the students’ discussions had stirred a genuine interest in Ariane**, an interest in this strange number, and she felt the need to study and write down everything that made 0 so special. She focused her study on three things:

*There are many weird things about 0:*

*0 : 0*

*0 ^{0}*

*0 itself*

**division as the inverse operation of multiplication**, an idea that we consolidate with students thanks to Multiplication Triangles. As 1 × 0 = 0; 2 × 0 = 0; 3 × 0 = 0… and any number multiplied by 0 is 0, then 0 ÷ 0 can have any number as an answer. Makes all the sense in the world, doesn’t it? The second operation she covers is the one that triggered everything: 0

^{0}. And look at Ariane’s wonderful argument to get to the answer,

**how she connects the operation she wants to decipher with the construction of any power with exponent 0 worked in the classroom**.

*To do this [the operation 0 ^{0}]you have to do 0 ÷ 0. Because 0^{1} = 0. To go from 0^{1} to 0^{0} you divide by 0, then [0^{0}] gives the same answer as 0 : 0.*

*Zero is a very strange number. In Roman numerals there is no 0. But even if there was 0, it is a number, but it is ‘nothing’. Zero represents nothing.*

## Knocking on the Door of Math

Ariane ends her study with a final, exciting conclusion. Exciting because she conveys that she has understood what math is and what it means to do math. And she was encouraged to investigate this world.

*I come to this world to bother people!*

*Until now I thought that math was ‘predictable’. But now I know that it is not all “look, this means that”, it’s that because they are odd, or because if you multiply… But not everything in math makes sense. And this is because of a number. This number is 0. Math would be very boring without 0.*

As you can see, to do relevant mathematical research, you don’t need to progress in the content from higher grades: you can do sophisticated mathematics with elementary content. With the content corresponding to Ariane’s level — that is, without peeking at content that will surely awe her in the future — we have been able to provide her with an interesting line of research. **An 11-year-old feeling called upon to do math of her own accord is a fact we cannot ignore**. A fact that we have to value very highly. And now we have a great responsibility: we have to try to keep this fire that has been lit burning, a fire that illuminates our own lives a little more.