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

Notation: We usually write a - b instead of a + (-b).

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.

    ________________________________________________
                0           1

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 a > 0. If a and b are two points on the number line such that a - b > 0, then we say a is greater than b and denote this by a > b. We may also write b < a if this is more convenient. That is, a > b and b < a mean exactly the same thing. You recognize it on the number line as a being to the right of b. We read b < a as "b is less than a".

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


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