Slide 1
... In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.[1]Since q may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or bla ...
... In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.[1]Since q may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or bla ...
Solutions
... ##NULL then build as you go or build space ahead of time ##We're going to do one of each here ##Initializing the median.vector variable (to build as we go) median.vector<-NULL ##Build space for the matrix ahead of time norm.matrix<-matrix(NA,n,p) ##Now filling in the matrix row-by-row ##each row has ...
... ##NULL then build as you go or build space ahead of time ##We're going to do one of each here ##Initializing the median.vector variable (to build as we go) median.vector<-NULL ##Build space for the matrix ahead of time norm.matrix<-matrix(NA,n,p) ##Now filling in the matrix row-by-row ##each row has ...
Integers – all whole numbers and their opposites
... Integers – all whole numbers and their opposites. Real-world examples of integers: Elevations: Above sea level (such as Mount Rainier in Washington at 14, 410 ft) – positive integers Below sea level (such as Death Valley, CA at 282 ft) – negative ...
... Integers – all whole numbers and their opposites. Real-world examples of integers: Elevations: Above sea level (such as Mount Rainier in Washington at 14, 410 ft) – positive integers Below sea level (such as Death Valley, CA at 282 ft) – negative ...
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.