Real Analysis, Lecture 2: Properties of Q
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when explaining equivalence classes and how fractions are constructed, and when working through the course's own exercises.
Recommended by 2 people across 2 different months, 2014 to 2017.
What it is
A university lecture from Professor Francis Su's Spring 2010 Real Analysis course at Harvey Mudd College. It covers properties of the rational numbers Q, including how fractions are defined as equivalence classes of ordered pairs of integers.
What people said
Recommends Juan Sebastian Lozano · 2014-04 · Mathematics Stack Exchange · on “Equivalence Class.”
As this [this video] lecture explains, an equvalance class is just a set of objects that are equal. Let us start by defining equality: we say that a relation is an equality (denoted by '$=$' usually, but can be arbitrarily represented), if and only if it is: reflexive : this simply means the an object is equal to itself (e.g. $a =a$, we obviously need this to be true, or else we could say things like $1 \not = 1$ !) Symetric : this means that if $a = b $ then $b = a$ (again this should be obvious because if not, then we could say things like $2 = 1 + 1$ and at the same time say $1 + 1 \not = 2$.) Transitive : this means that if $a = b$ and $b = c $ then $a = c$ ( if this were not true then i coul
Recommends EthanAlvaree · 2014-10 · Mathematics Stack Exchange · on “Multiplying and adding fractions”
As Sami hints, we define fractions in terms of equivalence classes of ordered pairs. You may find this lecture on properties of $\mathbb{Q}$ helpful; I certainly did at one point: [this video]
Cites supremus_01 · 2017-02 · Mathematics Stack Exchange · on “Proving positive quotients are well defined”
I'm going through the Harvey Mudd Real Analysis lectures. In the second lecture [this video] the professor gives a little problem to check knowledge: if $m/n$ is a positive quotient, then $mn > 0$ How would you show that this is well defined? I understand that if $m/n = p/q$ and $mn > 0$ therefore $pq > 0$ but am unsure where to proceed from there.