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# 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.