Lecture 1: Introduction to complex algebra
... monotonic decreasing sequence of real numbers must either tend to −∞ or to a finite real number. The set of all rational numbers form an ordered field, but is not complete. This means that the limit of a sequence of rational numbers need not be a rational number. Cauchy and Dedekind showed that the ...
... monotonic decreasing sequence of real numbers must either tend to −∞ or to a finite real number. The set of all rational numbers form an ordered field, but is not complete. This means that the limit of a sequence of rational numbers need not be a rational number. Cauchy and Dedekind showed that the ...
document
... the segments. The segments from the origin to the fourth vertex of the parallelogram represents the sum of the two original numbers. ...
... the segments. The segments from the origin to the fourth vertex of the parallelogram represents the sum of the two original numbers. ...
Chapter 9- Fibonacci Numbers Example: Rabbit Growth Start with 1
... Example: F1 = 1, F2 = 1, F3 = 2, and F4 = 3 A recursive definition for the Fibonacci Numbers: Let F1 = 1 and F2 = 1 be the starting values (seeds) for the sequence. FN +1 = FN −1 + FN for N ≥ 3 Try it with the Fibonacci Sequence: ...
... Example: F1 = 1, F2 = 1, F3 = 2, and F4 = 3 A recursive definition for the Fibonacci Numbers: Let F1 = 1 and F2 = 1 be the starting values (seeds) for the sequence. FN +1 = FN −1 + FN for N ≥ 3 Try it with the Fibonacci Sequence: ...
is a real number and
... This is referred to as the Null Set or Empty Set. Caution: Do not write the {0} set as the null set. This set contains one element, the number 0. Example 5: To show that 3 “is a element of” the set {1,2,3}, use the notation: 3 {1,2,3}. Note: This is also true: 3 N Example 6: 0 N where is rea ...
... This is referred to as the Null Set or Empty Set. Caution: Do not write the {0} set as the null set. This set contains one element, the number 0. Example 5: To show that 3 “is a element of” the set {1,2,3}, use the notation: 3 {1,2,3}. Note: This is also true: 3 N Example 6: 0 N where is rea ...
File
... Divisibility of Numbers: •How do we know if a number is divisible by 2? •It ends in 0, 2, 4, 6, or 8 •How do we know if a number is divisible by 3? •Add the digits. If the sum is divisible by 3, then the main number is. Ex: 1,278? Ex: 5,773? •How do we know if a number is divisible by 4? •If it end ...
... Divisibility of Numbers: •How do we know if a number is divisible by 2? •It ends in 0, 2, 4, 6, or 8 •How do we know if a number is divisible by 3? •Add the digits. If the sum is divisible by 3, then the main number is. Ex: 1,278? Ex: 5,773? •How do we know if a number is divisible by 4? •If it end ...
Turing Machines
... Decidable Implies Enumerable •Let L be a language over the alphabet •If L is decidable there is a Turing Machine that decides M •Let M* be the Turing machine that enumerates * •Construct a 3-tape Turing machine that enumerates L as follows: M* will output words on the 2nd tape Every time M* o ...
... Decidable Implies Enumerable •Let L be a language over the alphabet •If L is decidable there is a Turing Machine that decides M •Let M* be the Turing machine that enumerates * •Construct a 3-tape Turing machine that enumerates L as follows: M* will output words on the 2nd tape Every time M* o ...
