An irrational inequality

Go and think about it...Developing5.1

An irrational inequality

Excerpt: Students are asked to think about how we can prove that $e^\pi> \pi^e$ . How do we even define those quantities given that $e$ and $\pi$ are irrational?

Problem

It is astonishing that $e^\pi$ and $\pi^e$ are defined. It is even more amazing that we can prove $e^\pi> \pi^e.$

How would we go about convincing someone that this amazing inequality is true?

Suggestion

There are several ways to get a feel for this inequality, here are some suggestions:

what do the functions $e^x$ and $x^e$ look like?

how could we approximate $e$ and $\pi$ ?

do we know any functions that change a question about indices into a question about something else?

How do we define $e^\pi$ and $\pi^e$ given that both numbers are transcendental?

An irrational inequality

Go and think about it...Developing5.1

An irrational inequality

Excerpt: Students are asked to think about how we can prove that $e^\pi> \pi^e$ . How do we even define those quantities given that $e$ and $\pi$ are irrational?

Problem

It is astonishing that $e^\pi$ and $\pi^e$ are defined. It is even more amazing that we can prove $e^\pi> \pi^e.$

How would we go about convincing someone that this amazing inequality is true?

Suggestion

There are several ways to get a feel for this inequality, here are some suggestions:

what do the functions $e^x$ and $x^e$ look like?

how could we approximate $e$ and $\pi$ ?

do we know any functions that change a question about indices into a question about something else?

How do we define $e^\pi$ and $\pi^e$ given that both numbers are transcendental?