Download The Real Numbers Sequences are functions over the natural

yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Approximations of π wikipedia, lookup

Ethnomathematics wikipedia, lookup

Law of large numbers wikipedia, lookup

History of logarithms wikipedia, lookup

Location arithmetic wikipedia, lookup

Foundations of mathematics wikipedia, lookup

Infinity wikipedia, lookup

Georg Cantor's first set theory article wikipedia, lookup

Surreal number wikipedia, lookup

Non-standard analysis wikipedia, lookup

Proofs of Fermat's little theorem wikipedia, lookup

Infinitesimal wikipedia, lookup

Mathematics of radio engineering wikipedia, lookup

Positional notation wikipedia, lookup

Large numbers wikipedia, lookup

Arithmetic wikipedia, lookup

Hyperreal number wikipedia, lookup

Number wikipedia, lookup

P-adic number wikipedia, lookup

Real number wikipedia, lookup

Elementary mathematics wikipedia, lookup

Addition wikipedia, lookup

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?
Rational Numbers
Definition: Q = { } where a and b are integers and b 6= 0
Question: How many rational numbers are there between 0 and 1?
Back to f (q) = 1 + 2q
Irrational Numbers
Definition: The irrational numbers are numbers that cannot be expressed as where a and b are
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
+ d2 2 + d3 3 + d4 4 + ....
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.