Numbers

- This course will deal with what are commonly called "Real Numbers". The figure at the right shows the divisions of real numbers.
Real Numbers
• At the most basic level are Natural Numbers. The set of natural numbers includes all of the positive whole numbers that you have seen throughout your life, 1, 2, 3, 4, ... In set notation this is written as {1, 2, 3, 4, 5, 6, 7, ....} Note that this set does not include zero.
• If you add zero you have the set of Whole Numbers, {0, 1, 2, 3, 4, 5, 6, .....}
• The next step is to define Integers which includes both positive and negative numbers as well as 0. The set of integers can be show as, {... - 4, -3, -2, -1, 0, 1, 2, 3, 4....}. Positive and negative numbers are called signed numbers.
• Up to this point we have only dealt with integer values, the next step is to introduce Rational numbers. These numbers are the quotiennt of two integer numbers as long as the number that is the denominator is not 0. Representatives of this set of numbers includes, -3/7, - 2/4, +1/4, + 7/9 etc... Real numbers can be written as decimals. When you write them that way there are two possibilities.
Rational Real numbers as decimals
1. Terminating rational numbers - The decimal comes to an end, terminates at some point examples are 1/2 = 0.5, 1/4 = 0.25, 1/8 = 0.125
2. Repeating rational numbers - The decimal does not end but repeats forever. Examples are 1/3 = 0.3333......, 2/3 = 0.6666666, 3/11 = 0.2727272727.....
• Not all numbers are Rational. There is a set of numbers, called Irrational Numbers, that does not come from the ratio of two integers. √2 and √3 are examples as is π.
• For completeness we should mention numbers that are not real. These are called complex numbers and involve the square root of a negative number such as √-1. If you continue on in math you will encounter these numbers.

Graphing Numbers - At this point it is time to consider how to graph a real number on the number line. Think of the number line as a horizontal line with a scale marked on it. Various values occur to the right and left of the zero point of the line. Specifically positive numbers are found to the right of 0 and negative numbers are found to the left of 0. All of the real numbers discussed above can be graphed on the numberline. For example the number 2 would be 2 units to the right of 0. The number -3 would be 3 units to the left of 0 while 1.6 would be 1 and 0.6 units to the right of zero. The figures to the right showthe numbers just discussed graphed on the number line. The graphed point is indicated by a black circle and to assist you I have added an arrow and the number that is being graphed. Almost all textbooks show a horizontal number line and then at some later point show you a coordinate system with an x axis (horizontal line) a y axis (vertical line). These can be thought of as a horizontal and a vertical number line that cross at the zero point for both lines. To prepare you for the later introduction of a coordinate system we will next take a brief look at a vetical number line.

A Vertical Number Line - The numberline we just looked at was horizontal. Just for a minute let's look at a vertical numberline. These is no reason we can't have a numberline that is vertical. On this numberline 0 will again be in the middle with positive values above 0 and negative values below 0. The figure at the right shows three numbers graphed on a vertical number line. The first number, -2 is two units below zero. The second number, +3 is three units above zero and the third number, - 1.6 is 1.6 units below zero. Graphing numbers on the vertical numberline is just the same as graphing numbers on the horizontal number line except now the number line goes up and down instead of left and right.

Absolute Value - The absolute value of a number measures the distance of that number from zero. It is usually written as |number| such as |6| or |-4|. The number appears between two vertical lines. The absolute value of a number is always positive. So the absolute value of -3.25, written as |-3.25| is 3.25. The absolute value of +6.18, written as |6.18| is 6.18. If there is a sign outside the absolute value sign you apply it after you have evaluated the terms inside the absolute value. So -3 * | -4| = -3 * 4 = -12. The absolute value of -4 is 4. 4 times -3 is -12. The table shown below gives some problems you can use to test yourself on these ideas. Look at the problem, enter your answer, then click on answer and the correct answer will appear in the rightmost box.

Calculation of Absolute Values

Graphing Inequalities - Up to this point we have considering the classification of numbers and how to graph any given number when it is equal to a particular value. There is also a mathmatical relationship known as an inequality. These are as follows:

Inequalities
• Greater than - symbol > - example: x > 7
• Less than - symbol < - example: x < -2
• Greater than or equal to - symbol > - example x > -1
• Less than or equal to - symbol < - example: x < 6.17

Let's look at some inequalities and graph them. The first one we will look at is x > 12. In words this says the x is greater than 12. If we go to the number line this means that x can be any number that is greater than 12. So x can be 12.0001 or 13 or 2563. On a graph we indicate the values the variable can take on with an open or closed circle at the starting point and an arrow in the direction of acceptable values. The figure on the right shows the graph for this case.

Now let's look a a similar problem where x is greater than or equal to 12. This is written as x > 12. Since the equals condition is present we will used a filled circle at the starting point to show that x can take on the value of 12. In the problem before this we used an open circle to indicate that x could not take on that value.

In the previous problems x was greater than a given value so let's now look at the case where x is less than some value. Suppose we are told that x is less than -3. This can be written as x < -3 and the graph is shown to the right. We have used an open circle at the point where x is -3 to indicate that x can be very close to -3 but cannot take on that value. x must be less than -3. The arrow goes to the left, those are the values that are less than -3. For example -4, -5 and -6 are all less than -3.

Finally let's look at the case where x is less than or equal to 4. One way to write this is x < 4. The graph is shown to the right. We have used a filled in circle at x = 4 to indicate that x can take on that value. The arrow starts at 4 and goes to the left since x is less than (to the right of) 4. So +2, -5 and -6 are all acceptable values of x.