Download The Real Numbers Sequences are functions over the natural

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Transcript
The Real Numbers
Sequences are functions over the natural numbers. Consider the sequence generated by
f (n) := 3 + (n − 1)2
We now want to transition to talking about functions defined over the real numbers. But.... what
are the real numbers?
Integers
Rational Numbers
a
Definition: Q = { } where a and b are integers and b 6= 0
b
Question: How many rational numbers are there between 0 and 1?
Back to f (q) = 1 + 2q
Irrational Numbers
a
Definition: The irrational numbers are numbers that cannot be expressed as where a and b are
b
integers and b 6= 0.
How are the real numbers related to sequences/series? Let’s consider decimal expressions.
Example 1:
a) 7134
b) 60.524
c) 58.722222.....
d) 4.13636363636363636.....
e) 4.72195236....
So in general, every decimal can be expressed as a series (either finite or infinite). Let the following
represent any decimal
z . d1 d2 d3 d4 ...
This can be expressed as
z + d1
1
1
1
1
+ d2 2 + d3 3 + d4 4 + ....
10
10
10
10
How do these decimal expressions relate to the types of numbers we defined above?
Example 2:
a) 60.524
b) 4.136
As these examples indicate, every terminating or repeating decimal is a rational number.
Non-repeating, non-terminating decimals are also real numbers. Since they are not rational, by
definition they must be irrational.
The point: The real numbers are completely made up of the rational numbers and the irrational
numbers. The real line is continuous (no holes or gaps of any kind).
Example 5:
f (x) := 1 + 2x where x is a real number.
All of the basic functions you have ever dealt with are defined over the real numbers or some subset
of the real numbers:
These functions are all continuous on their domain. We will re-examine each of these functions
using sequences, limits, and continuity to better understand their behavior.