Basic Rules of Algebra
The real numbers form an abstract set, denoted R which contains two special distinct numbers 0 and 1 and for which we have two operations, addition and multiplication that combine two given real numbers to make a third real number. Addition of a and b is denoted a+b and multiplication of a and b is denoted ab. Moreover, the following basic rules are satisfied:
 BR1

a + b = b + a , for all a, b in R  BR2

(a+b) + c = a + (b+c) , for all a, b, c in R  BR3

a + 0 = a , for all a in R  BR4
 For any a in R, there exists a number a in R so that
a + (a) = 0  BR5
 ab = ba , for all a, b in R
 BR6
 (ab)c = a(bc) , for all a, b, c in R
 BR7
 a1 = a , for all a in R
 BR8
 If a in R and a not equal to 0, then there exists a number b in
R so that ba = 1 (Usually this b is denoted 1/a)
 BR9

a(b+c) = ab + ac , for all a, b, c in R
This rule is called expanding. If you are given an expression of the form (a + b)(c + d) and are told to expand it, you essentially do the same thing. Informally, it is called "FOILing" the expression. "FOIL" stands for First, Outside, Inside and Last. This means that you multiply the first term in each bracket together (i.e. ac), then multiply the outside terms together (i.e. ad), then multiply the inside terms together (i.e. bc) and finally multiply the last terms together (i.e. bd).The result is:
(a + b)(c + d) = ac + ad + bc + bd, for all a,b,c,d in R
The real numbers are usually pictured as a number line, which is a horizontal straight line with 0 and 1 marked on it with 1 to the right of 0.
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There are order properties that the real number line has that originate in the
choice of 1 being to the right of 0. If a is a point on the number
line on the same side of 0 as 1 is, we say a is greater than 0 and denote this by
There are three more basic rules of the real numbers that involve order. We list them here formally so that they can be easily refered to later.
 BR10
 For any a in R, exactly one of the following statements is true:
 (i)
 a > 0
 (ii)
 a = 0
 (iii)
 a < 0
 BR11

For a,b in R, if a > 0 and b > 0, then a + b > 0  BR12

For a,b in R, if a > 0 and b > 0, then ab > 0