By definition, exp(m) is the number of times that 2 divides m; this means that exp(m) is always a natural number. For example, exp(4)=2, exp(5)=0, exp(6)=1, exp(8)=3.

Hence, you can never find m and n, such that exp(m)=exp(n)+0.5, because the left-hand side is a natural number and the right-hand side is not.

I hope this helps,
Joao

Thanks. =)

that’s the contrapositive of Leibniz’s rule. Leibniz rule states that, for all a, b, and function f:
a=b => f.a = f.b

The contrapositive is obtained by negating both sides and changing the direction of the implication:

a=/=b <= f.a =/= f.b

Thanks for your comment,
Joao

Each step in the proof format I use has the form:

A
R { hint }
B .

This means that A is related with B by relation R, and the reason is shown in the hint. If I instantiate R with equality, I get:

A
= { hint }
B ,

which means that A=B.

Similarly, if I use the arrow, as in

A
< = {hint}
B ,

then it means that B implies A.

***

In this particular example, I am using the contrapositive of the so-called Leibniz rule:

f(a) =/= f(b) => a =/= b .

In words, if the value of f(a) is different from f(b), then a must be different from b.

I hope this helps,
Joao

However, I have a small question from my ignorance on your notation.

In-between the statements, there are lines that explain the transformation. All of them start with an equal sign except the second one, which is left-arrow. What does that mean and why is this the only one?

The so-called Leibniz rule (also called ‘substitution of equals by equals’) states that, for all a, b, and function f:

a = b => f.a = f.b .

Hence, you can see the first step as a mutual implication where, in one direction, function f is taking the square root, and in the other direction, function f corresponds to squaring.

Thanks for your comment!

delicious!truly elegant!

isn’t the first equivalence actually an implication? m/n could be minus sqrt(2)…

tks

of course I have. It was a typo, I’m sorry. It is corrected now. Thanks a lot!