Fractions
... Convert the improper fraction, 7/5 to a mixed number. What do we do first? Write 7/5 as a division expression! Now what? Solve the expression! Finally… Use the quotient as the whole number, the remainder as the numerator, and the divisor as the denominator! ...
... Convert the improper fraction, 7/5 to a mixed number. What do we do first? Write 7/5 as a division expression! Now what? Solve the expression! Finally… Use the quotient as the whole number, the remainder as the numerator, and the divisor as the denominator! ...
A Musician`s Guide to Prime Numbers
... of congruences. Before we do so, let us first try to compile the musical conjectures into one mathematical conjecture. Conjecture 1.4. There are infinitely many prime numbers of the form 12k +r, for some k ∈ N and r ∈ R = {1, 5, 7, 11}. For any positive integer n ≤ 12 with n∈ / R, 12k + n is never p ...
... of congruences. Before we do so, let us first try to compile the musical conjectures into one mathematical conjecture. Conjecture 1.4. There are infinitely many prime numbers of the form 12k +r, for some k ∈ N and r ∈ R = {1, 5, 7, 11}. For any positive integer n ≤ 12 with n∈ / R, 12k + n is never p ...
The Number Of Certain k-Combinations Of An n-Set
... In looking for the possible coalitions among n persons, we allow for a partial set partition of the n-set S instead of a complete partition. This circumstance leads to the observation that the number of possible coalitions (disjoint strict k-combinations) among n persons is B(n + 1) instead of B(n) ...
... In looking for the possible coalitions among n persons, we allow for a partial set partition of the n-set S instead of a complete partition. This circumstance leads to the observation that the number of possible coalitions (disjoint strict k-combinations) among n persons is B(n + 1) instead of B(n) ...
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.