Welcome to your MAT 1375 class site on OpenLab.
If you have not already got a copy of the textbook, it is available here.
The topics for today:
• the Real Number system. It is background to what we do in the rest of the chapter (and the rest of the course!)
First we should understand very clearly that certain words used in Mathematics (or any technical subject) may be the same as words used in everyday talk, but in Mathematics these words have specific meanings which are often different from (but sometimes related to) the meanings that they have in everyday talk. In this case, “real” is just a name for this set of numbers: they are not more “real” in the everyday sense than any other set of numbers. And also, “number” in everyday talk almost always means what is called in mathematics a “natural number”. So we have to be careful that we use the words appropriate to the context!
The sets of numbers we considered: (see in the textbook for notation)
• The natural numbers (also called the counting numbers) 1, 2, 3, 4, …
• The whole numbers – these are not mentioned in Session 1 in the textbook, but it is a name we use a lot so you should know what they are. The whole numbers are just the natural numbers along with 0: 0, 1, 2, 3, 4, …
Be aware that some people use “whole number” as a synonym for “natural number” though!
• The rational numbers: please note that these are numbers which can be expressed as a ratio (or quotient) of two integers: they do not have to come to you already in this form. Thus 33 is a rational number because you can write it as $latex \frac{33}{1}$ (for example, or $latex \frac{66}{2}$, or $latex \frac{-66}{-2}$, etc.)
We mentioned that the denominator cannot be 0: the fact that division by 0 is impossible is rather important, and you can benefit by thinking carefully about it. In a sense, all of Calculus is concerned with this question!
• The real numbers: they are the points on the number line, interpreted as directed distances away from 0. Once you have designated a point to be 0 and another point to the right of it to be 1, that determines your unit of measurement and you can mark off all the other points using it (or fractions of it: it is a little mysterious how the irrational numbers are to be found though!) By saying “directed distances” I mean that the distance is counted as positive if the number is to the right of 0, and negative if the number is to the left of 0. (Normally distances are always positive.) This is the first instance of something we will later be calling vectors, which are lengths which also have direction.
• We noted that there are points (lengths) on the number line which are not rational numbers. The most famous example of this is the square root of 2. This is most definitely a length, since it is the length of the hypotenuse of a right triangle whose sides are both 1. (Use the Pythagorean Theorem and prove this to yourself!) However, the square root of 2 cannot be expressed as a ratio of integers. You can read the proof here.
There are many other real numbers which are not rational numbers. For example, pi, e (the base of the natural logarithms), and the square roots of any number which is not a perfect square, are all irrational. In fact, there are more irrational numbers than rational numbers. Weird!
• Inequalities and intervals and interval notation
• The absolute value of a real number. There are two ways of defining the absolute value: one is algebraic, and one is geometric.
Algebraic definition: for any real number $latex c$,
$latex |c| = c$ if $latex c \ge 0$,
$latex |c| = -c$ if $latex c < 0$.
(This defines the absolute value as a piecewise-defined function: we will return to these in a later Session!)
Geometric definition: $latex |c|$ is the distance from 0 to $latex c$ on the number line. (A distance is always non-negative.)
• Solving absolute value equations (see examples 1.4 – 1.9 in the textbook): notice that the first step is always to get rid of the absolute value sign by splitting the equation into two. (This is if the right-hand side is a positive number: what if the right-hand side is 0 or negative?)
• Solving absolute value inequalities by the “Test Point method”
(video at the link)
We only used the first method given in the textbook for solving the inequalities. (See Example 1.17) This method can also be used for other types of inequalities, and we will be using it later on. Please do not skip any of the steps!
Homework:
• Reread/review the material covered in class. This is most of Session 1 in the textbook.
• Add to or change your email address in CUNYFirst to an address that you usually check.
• If you do not regularly check your City Tech email, here are instructions about forwarding it to another email address.
• Log in to WeBWorK following the instructions here and start working on the assignments.
• Routine homework: in Session 1, Exercises 1.1, 1.2, 1.3, and 1.4 (the assigned parts: see the Course Outline), and 1.6 (all parts), 1.7 (a)-(f) – we did some of these in class. Any that we have not already put on the board, you may write all your work on the board next time as part of the 10 problems you need to put on the board for your grade. (Only problems that have work to show are to be put on the board!)
• If you have questions about the homework problems or anything else related to this material you can post them as questions on Piazza. Please check your email (whatever email address is in CUNYfirst for you) for the invitation to join Piazza.
