Normal numbers and the Borel hierarchy
... Now suppose ϕ is false. Let x be such that there are infinitely many y such that C(x, y). Let z be any positive integer. Each time an appending tuple of the form hx, yi with z < y is processed, x + z is appended to the output. Since we assumed there are infinitely many such tuples, x + z is appended ...
... Now suppose ϕ is false. Let x be such that there are infinitely many y such that C(x, y). Let z be any positive integer. Each time an appending tuple of the form hx, yi with z < y is processed, x + z is appended to the output. Since we assumed there are infinitely many such tuples, x + z is appended ...
More on Proofs – Part III of Hammack
... From Chapter 4 until the present time, we have dealt with proving that a given statement is true. Now, we will discuss what we should do if we are given a statement that is false. In order to prove that a statement is false, we carry out a procedure called disproof. This chapter, then, is concerned ...
... From Chapter 4 until the present time, we have dealt with proving that a given statement is true. Now, we will discuss what we should do if we are given a statement that is false. In order to prove that a statement is false, we carry out a procedure called disproof. This chapter, then, is concerned ...
Lecture #13 - February 27, 2004 - Introduction to Floating Point
... earlier example we saw that the decimal number 23.375 is 10111.011 in binary. The normalized scientific notation version of 23.375 is 2.3375×101, and the normalized binary version of 10111.011 is 1.0111011×24. With the exception of zero, every normalized binary number is of the form 1.xxxx×2YYYY, wh ...
... earlier example we saw that the decimal number 23.375 is 10111.011 in binary. The normalized scientific notation version of 23.375 is 2.3375×101, and the normalized binary version of 10111.011 is 1.0111011×24. With the exception of zero, every normalized binary number is of the form 1.xxxx×2YYYY, wh ...
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.