数式ミステリー㉞『数学者のメモ』Page5

The problem says, “Find the largest natural number.”
Here, natural numbers are $1,  2,  3,\cdots$.

"Hmm? The largest natural number?
That sounds strange.
There are infinitely many natural numbers.
So there is no largest natural number, right?
But don’t worry.
Even junior high school students can understand this.
Let’s check this together.

Let the largest natural number be $x$.
Then $x^2$ is the same as $x$, or $x^2$ is always smaller than $x$.
That is,  $x^2=x$  or  $x^2< x$.
So we get

$$x^2 \leqq x$$

Move $x$ to the left side.

$$x^2 − x \leqq 0$$

Solve this.
We change the left-hand side.
That is, we factor it like this.

$$x(x − 1)\leqq 0$$

First, since $x$ is a natural number, we have

$$x>0　\cdots(*1)$$

Next, we know  $x(x − 1)\leqq 0$.
If  $x(x − 1)\leqq 0$  and  $x>0$, then we must have  $x-1\leqq0$.
From  $x-1\leqq0$, we have

$$x\leqq1　\cdots(*2)$$

Using $(*1)$ and $(*2)$, we get

$$0 < x \leqq1$$

Look, the largest natural number that satisfies this is $x=1$.

What?
The largest natural number is $1$?

Wait, this is strange.
$1$ is not the largest natural number.
Strange, strange, strange!

Let us do the calculation one more time.

$x$ is the largest natural number.
　$x^2 \leqq x$
Move $x$ to the left-hand side.
　$x^2 − x \leqq 0$
Factor it.
　$x(x − 1)\leqq 0$
Solve this, and we get
　$0< x \leqq1$

$x$ is the largest natural number that satisfies this.
So the largest natural number is $1$!

A great discovery!
The largest natural number is $1$!!!

I am a genius.
I have found an amazing fact!
I will write a paper.
I will send it to a math society!!!

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