Real Numbers

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

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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                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 and b are two points on the number line with a to the right of b 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. 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