Direct proofs

The concept of direct proof can vary slightly from person to person. For instance, Wikipedia defines it as:

In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions.

Alternatively, in Larry W. Cusick’s website we can read:

A direct poof [sic] should be thought of as a flow of implications beginning with “P” and ending with “Q”.

P -> … -> Q

Most proofs are (and should be) direct proofs. Always try direct proof first, unless you have a good reason not to.

I consider the wording ‘without making any further assumptions‘ in the first definition ambiguous and I don’t understand why the second definition only applies to implications. But anyway, with these definitions in mind, a direct proof for the irrationality of can be something like:

Or, alternatively, we can also use a proof of the following shape:

Irrational numbers

An irrational number is a real number that can’t be expressed as a simple fraction. Therefore, the number is irrational because for all integers m and n, with n non-negative, we have that:

A direct proof for the irrationality of

Now that we have clarified the concepts above, we prove that is irrational. For all integers m and n, with n non-negative, we have:

Note that, unlike traditional proofs, we don’t assume that m and n are co-prime, nor that is a rational. We essentially derive the boolean value of the expression

If you have any suggestions or corrections, please leave a comment. I’d be more than happy to hear from you.

Note: I learnt the contrapositive of this proof from Roland Backhouse (page 38, Program Construction — Calculating Implementations from Specifications).