# 06-240/Classnotes For Tuesday, September 12

• PDF notes by User:Harbansb: September 12 Notes.
• If I have made an error in my notes, or you would like the editable OpenOffice file, feel free to e-mail me at harbansb@msn.com.
• PDF notes by User:Alla: Week 1 Lecture 1 notes
• Below are a couple of lemmata critical to the derivation we did in class - the Professor left this little work to the students:

# Notes

## The Real Numbers

The Real Numbers are a set (denoted by ${\displaystyle \mathbb {R} }$) along with two binary operations: + (plus) and · (times) and two special elements: 0 (zero) and 1 (one), such that the following laws hold true:

${\displaystyle \mathbb {R} 1}$: ${\displaystyle \forall a,b\in \mathbb {R} }$ we have ${\displaystyle a+b=b+a}$ and ${\displaystyle a\cdot b=b\cdot a}$ (The Commutative Laws)

${\displaystyle \mathbb {R} 2}$: ${\displaystyle \forall a,b,c\in \mathbb {R} }$ we have ${\displaystyle (a+b)+c=a+(b+c)}$ and ${\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}$ (The Associative Laws)

${\displaystyle \mathbb {R} 3}$: ${\displaystyle 0}$ is an additive unit and ${\displaystyle 1}$ is a multiplicative unit (The Existence of Units/Identities)

${\displaystyle \mathbb {R} 4}$: ${\displaystyle \forall a\in \mathbb {R} \ \exists b\in \mathbb {R} {\mbox{ s.t.}}\ a+b=0}$

This is incomplete.