Lecture 2: Classification of Functions
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Reminder: The class will meet in room 420 of Fretwell
on Friday, January 16. A representative of computer services
will help you access your Eudora account. Please bring a 3 1/2
diskette to this class.
Brief review of the lecture
This is a summary of the second lecture in Math 1100 last semester,
left here because it is relevant to this course. We will spend very little time on this
material since you have all studied this.
Preliminary Stuff. There are three important types of objects in our
study of algebra: 1. Numbers, 2. Algebraic Expressions, and 3. Functions.
By numbers we mean real numbers. We need to understand how to represent them,
how to convert from one representation to another, and how to do
arithmetic-that is, how to combine numbers using the arithmetic operations
plus, minus, times, divide, exponentiate, etc.
We also learn about ordering numbers.
We learn rules of arithmetic and order. This is what chapter
P is about. The next few chapters do the same thing with algrebraic
expressions that the first one does with numbers. There are a few more
things we can do with these expressions than we can do with numbers.
For example, we can evaluate them. For example, the expression
4xy^2 has the value 3 when x=3 and y=1/2.
The third part of our study of algebra will involve functions. A function
is a relationship between two sets of numbers. Functions are ofter defined
using algebraic expressions. Functions from the real numbers to
the real numbers have graphs which we use to understand the
`behavior' of the function.
1. The need for numbers. Our number systems have grown as the need for
new numbers grew. Cave men had only two numbers, few and many.
Irrational numbers were `invented' to solve equations like
x times x-2=0.
2. Exponents and radicals. The meaning of exponential notation.
3. The need to distinguish two aspects of number. Each
(real) number has a value, and a representation (in fact each number has many
representations). When we say two numbers are equal, we mean they
have the same value.
4. The real number system. This will have to wait until the third
lecture.
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