Introduction to, or Review of, Series The absolute value of a complex
... has been turned upside down, because it has been moved to the other side of the inequality. One easy way to remember which way the fraction goes is by looking at a simple example, such as z z2 z3 z4 ...
... has been turned upside down, because it has been moved to the other side of the inequality. One easy way to remember which way the fraction goes is by looking at a simple example, such as z z2 z3 z4 ...
1 The Natural Numbers
... approaches the non-rational number 2, a fact well known since antiquity. We want to remedy this deficiency: we want to construct an ordered field F containing the rational numbers, which is “complete” in the following sense: (C1) Every increasing13 bounded14 sequence15 of elements in F converges16 t ...
... approaches the non-rational number 2, a fact well known since antiquity. We want to remedy this deficiency: we want to construct an ordered field F containing the rational numbers, which is “complete” in the following sense: (C1) Every increasing13 bounded14 sequence15 of elements in F converges16 t ...
Absolute Value of an Integer
... For any two different places on the number line, the integer on the right is greater than the integer on the left. Examples: 9 > 4, 6 > -9, -2 > -8, and 0 > -5 -2 < 1, 8 < 11, -7 < -5, and -10 < 0 Absolute Value of an Integer The absolute value of a number is always a positive number (or zero). We s ...
... For any two different places on the number line, the integer on the right is greater than the integer on the left. Examples: 9 > 4, 6 > -9, -2 > -8, and 0 > -5 -2 < 1, 8 < 11, -7 < -5, and -10 < 0 Absolute Value of an Integer The absolute value of a number is always a positive number (or zero). We s ...
Full text
... Solution by Leonard A. G. Dresel, Reading, England We shall find two solutions, namely a = 6, b = 9, c = 4, and a - 12, b - 9, c - 72, and show that these are the only solutions. Putting n = b in the given identity, we have Fb = HFb_a, so that 17 divides i^,, giving Z? = 9, 18, 27, ... . In fact, b ...
... Solution by Leonard A. G. Dresel, Reading, England We shall find two solutions, namely a = 6, b = 9, c = 4, and a - 12, b - 9, c - 72, and show that these are the only solutions. Putting n = b in the given identity, we have Fb = HFb_a, so that 17 divides i^,, giving Z? = 9, 18, 27, ... . In fact, b ...
Aalborg Universitet Aesthetics and quality of numbers using the primety measure
... investigate specific number categories with respect to their primety value. For instance, the pseudoprimes all have different primety values, although always above 0.5 for numbers up to l06. As a comparison, the minimum primety value in that range is below 0.2. The lucky numbers also have primety va ...
... investigate specific number categories with respect to their primety value. For instance, the pseudoprimes all have different primety values, although always above 0.5 for numbers up to l06. As a comparison, the minimum primety value in that range is below 0.2. The lucky numbers also have primety va ...
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.