Real Numbers

Hopefully most of this section will be an easy for you and will largely be a review.

First let's look at what is meant by real numbers.

Natural Numbers, what you might call positive integers, 1, 2, 3, 4 ...
Integers, including the negative and positive integers, ...- 3, -2, -1, 0 ,1, 2, 3, 4 ...
Rational Numbers, numbers formed by taking the ratio of integers, 3/2 = 1.5, 6/8 = 0.75
Irrational Numbers, numbers that cannot be formed from the ratio of two integers, square root of 2, p, etc...

Now let's consider some of the properties of real numbers.

Commutative, a + b = b + a, with numbers, 6 + 7 = 7 + 6
Associative (addition), (a + b) + c = a + (b + c), with numbers, (6 + 5) + 3 = 6 + ( 5 + 3)
Associative (multiplication), (ab)*c = a*(bc), with numbers, (2*3)*5 = 2*(3*5)
Distributive, a(b + c) = ab + bc, with numbers, 3( 2 + 6) = 3*2 + 3*6

What if we have negative numbers?

(-1)a = -a, with numbers, (-1)*8 = -8
-(-a) = a, with numbers, -(-9) = 9
(-a)b = a(-b) = -(ab), with numbers, (-2)*6 = -(2*6)
(-a)(-b) = ab, with numbers, (-7)(-4) = 7*4
-(a + b) = -a -b, with numbers, -(8 + 2) = -8 - 2
-(a - b) = b -a, with numbers, -(6 - 3) = 3 - 6

The hardest part of this section will probably be fractions which will be covered next.

(a/b) * (c/d) = (ac)/(bd), with numbers, (2/3) * (5/7) = (2*5)/(3*7) = 10/21
(a/b) ÷ (c/d) = (a/b) * (d/c) = (ad)/(bc), with numbers, (2/9) ÷ ( 5/7) = (2/9) * (7/5) = (2*7)/(9*5) = 14/45
(a/c) + (b/c) = (a + b)/c, with numbers, (2/5) + (8/5) = (2 + 8)/5 = 10/5 = 2 (common denminator is 5)
(a/b) + (c/d) = (ad + cb)/(bd), with numbers, (2/5) + (5/4) = (2)/(5) * (4/4) + (5/4) * (5/5) = (8/20) + (25/20) = 33/20

In this case we had to first determine the common denominator which was 5*4 or 20

To make the denominators the same we multiplied the first by 4 and the second by 5

But, if we just multiplied the denominators we changed the fraction, so we multiplied each numerator by the same
number.
Now we had common denominators and could add the fractions
.

(ac)/(bc) = a/b, with numbers, (2*9)(5*9) = 2/5
If (a/b) = (c/d) then ad = bc, with numbers, if (1/2) = (8/16) then 1*16 = 2*8 which is true (we cross multiplied here)

One way to represent real numbers is to locate them as points along a line. This is typically called the real number line of the coordinate line. On graphs that you have seen the x axis is a horizontal version of this line and the y axis is a vertical version of the same type of line. The figure shown below shows some points on the real number line. //Insert a figure here that shows the real number line

We need a way to tell others where points are along the line, or which points are of interest to us or fit a particular mathematical relationship. To do this your textbook uses set notation. Your textbook on page 9 in the blue box shows three ways to represent points.

Notation of the form (a,b) or [a, b) or various combinations. A ( means the point is not included while a [ includes the point

Thus the interval (2, 7) goes from 2 to 7 but does not include the end points

The interval [2, 7) also goes from 2 to 7 but it includes 2 (but not 7)

The interval [2, 7] includes both endpoints, 2 and 7

Set descriptions can be written as {x | a < x < b}, which is read as the set of x where x goes from a to b, not including the endpoints
The last technique uesd is to display a graph of the points. In this case an open circle is used to indicate that a point is not included and a closed circle or dot is used to indicate that a point is included. //add figure

We may combine sets in several ways. Your textbook treats intersections and unions. If we are treating two sets their intersection will be only the points that are common to both sets. Their union will be the set of points that are in either set.

Intersection of (2, 7) and (3,9) is (3, 7) because only the points from 3 to 7 are common to both sets (endpoints not included)
Union of (2, 7) and (3, 9) is (2, 9) because the points from 2 to 9 are in one of the two sets (endpoints not included)

The last things discussed in this section are absolute value and distance. The absolute value of a number tells you how far that number is from zero. In other words, it gives you the distance from zero, but without a sign. It does not tell you if the number is above or below zero.

|a|>= 0 (where > = means greater than or equal to), with numbers, |-9| = 9
|a| = |-a|, with numbers, |4| = |-4| both give the result, 4
|ab| = |a| |b|, with numbers, |3*5| = |3| |5| = 15
|a/b| = |a|/|b|, with numbers, |9/-3| = |9|/|-3| = 9/3 = 3