Is this proof "by contradiction" or "by contrapositive"?

[Hill]

TL;DR

The text claims that this is an example of "proof by contradiction", but it seems rather to be a "proof by contrapositive."

This is the example in question:

I think that this is rather an example of "proof by contrapositive": given a rational $a$, instead of proving directly that
$b \text{ irrational} \Rightarrow ab \text{ irrational}$
they have proved the contraposition,
$ab \text{ rational} \Rightarrow b \text{ rational}$.

They did not use in their proof an assumption of $b$ being irrational. Thus, they did not produce any contradiction.

 

[Hornbein]

TL;DR: The text claims that this is an example of "proof by contradiction", but it seems rather to be a "proof by contrapositive."

This is the example in question:

View attachment 371514

I think that this is rather an example of "proof by contrapositive": given a rational $a$, instead of proving directly that
$b \text{ irrational} \Rightarrow ab \text{ irrational}$
they have proved the contraposition,
$ab \text{ rational} \Rightarrow b \text{ rational}$.

They did not use in their proof an assumption of $b$ being irrational. Thus, they did not produce any contradiction.

The text is correct. They assume the theorem is false and show this leads to a contradiction of the condition that b is irrational.

 

[FactChecker]

they have proved the contraposition,
$ab \text{ rational} \Rightarrow b \text{ rational}$.

They did not use in their proof an assumption of $b$ being irrational. Thus, they did not produce any contradiction.

They did initially assume that a is rational and b is irrational, and they used the assumption about a in the proof to say that a=m/n. So this is more than just assuming that ab is rational.