ppt - School of Computer Science
... Let be any fixed finite set of symbols. is called an alphabet, or a set of symbols. Examples: ...
... Let be any fixed finite set of symbols. is called an alphabet, or a set of symbols. Examples: ...
7 Sequences of real numbers
... Is the converse of Theorem 7.3.2 true? The converse is: If a sequence is bounded, then it converges. Clearly a counterexample to the last implication is the sequence (−1)n , n ∈ N. This sequence is bounded but it is not convergent. The next question is whether boundedness and an additional property ...
... Is the converse of Theorem 7.3.2 true? The converse is: If a sequence is bounded, then it converges. Clearly a counterexample to the last implication is the sequence (−1)n , n ∈ N. This sequence is bounded but it is not convergent. The next question is whether boundedness and an additional property ...
Mixed Numbers and Improper Fractions
... Writing Improper Fractions as Mixed Numbers • If you have an improper fraction, you can divide the denominator into the numerator. • The quotient becomes the whole number part of the mixed number. • The remainder is the numerator of the fraction. • The divisor is the denominator of the fraction. ...
... Writing Improper Fractions as Mixed Numbers • If you have an improper fraction, you can divide the denominator into the numerator. • The quotient becomes the whole number part of the mixed number. • The remainder is the numerator of the fraction. • The divisor is the denominator of the fraction. ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.