Mathematics is coming and going?

If a, b belong R (a and b belongs to the group of the numbers real)

i) a < b    if and only if    b - a    is positive

ii) a > b    if and only if    a - b    is positive 

But in school is like this

If a, b belong R (a and b belongs to the group of the numbers real)

i) if a < b    so a - b is positive

ii) if a > b so a - b is negative  -  here is a consequence if you slip you fall.

I think the last one is telling about what is rational and the other is like went to China to conclude that there is talk Chinese, let me say backward and will remain true, but I am saying backward. Rute Bezerra de Menezes Gondim