Posts Tagged ‘n!’
Thursday, December 17th, 2009
The Story of the Infinite Hotel
Description
A detailed tale of the Infinite Hotel. Step by step story including several pictures and an explanation of the Infinite Hotel for reference.
Overview
The Infinite Hotel is a famous math story and puzzle that was thought of by David Hilbert, a German mathematician. Sometimes the Infinite Hotel is called Hilbert’s Paradox of the Grand Hotel. It states that if one person comes into the hotel and all the rooms are full, they can all move down one room and the person can then take the first room. If k number of people come into the hotel and all the rooms are full, everyone can move down k number of rooms to make room for the people that just arrived. And, if double the amount of people that are already there are looking for rooms, everyone in room n can move to room 2n, making room for all the new arrivals in the odd-numbered rooms. This example of the Infinite Hotel can be used in certain forms of mathematical induction, and also in set theory and studies dealing with infinite numbers.
Tags: algebra, arrivals, David Hilbert, double, down, German, grand, Hilbert, hotel, induction, infinite, infinity, k, move, n!, new, numbers, paradox, room, set, space, theory
Posted in Algebra | No Comments »
Friday, November 20th, 2009
How to Pick Variables
Description
A detailed tutorial on how to pick variables. Step by step tutorial including several examples of how to pick variables for reference.
Overview
Variables are letters picked to represent unknown values in expressions and equations. Usually they are lowercase, but they can be made uppercase. When trying to pick a variable, you must choose wisely. x is the most common variable, followed by n. x is picked because people associate it with the unknown, and n is picked because it stands for “number.” The variable should be easily recognizable – you should not use a variable that looks like another number or some symbol of a mathematical operation. You should check to see what is included in your equation – for instance, m stands for slope, so if you are doing an equation with slope you need to pick a different variable to avoid confusion. And you should always pick a variable that makes sense – the first letter of your subject matter usually works quite well.
Tags: a, algebra, b, c, choose, equation, expression, lowercase, m, mathematical, n!, number, operation, slope, symbol, unknown, uppercase, value, variable, variables, x, y, z
Posted in Algebra | No Comments »
Thursday, November 12th, 2009
An Overview of Magic Squares
Description
A detailed tutorial of magic squares. Step by step tutorial including several examples of magic squares for reference.
Overview
Magic squares are a fun mathematical trick and puzzle. It is an arrangement such as 3×3, 4×4, or any other nxn pattern of numbers. Typically a magic square will contain any of the integers between 1 and n^2. Magic squares are set up so that all rows and columns, and both diagonals, add up to the same constant. It does not matter what constant it is, as long as all rows, columns, and diagonals add up to the same one.
Tags: arithmetic, column, constant, diagonal, integer, magic, n!, normal, number, perfect, real, row, square, sum, word
Posted in Arithmetic | No Comments »
Tuesday, November 3rd, 2009
Introduction to Square Matrices
Description
A detailed tutorial on square matrices and how to identify them. Step by step tutorial including several examples of square matrices for reference.
Overview
A square matrix is a simple matrix in the shape of a square. It has the same number of rows and columns. Square matrices are called nxn matrces. The most common values for n are 2 and 3. Two columns and rows is the smallest amount of rows and columns a square matrix can have – matrices with only one value are not considered to be square.
Tags: 2, 2x2, 3, 3x3, algebra, columns, equal, equivalent, linear, matrices, matrix, n!, number, nxn, rows, same, shape, square, three, two, values
Posted in Algebra | No Comments »
Tuesday, November 3rd, 2009
Well-Ordering Principle Explained
Description
A detailed tutorial on the well-ordering principle. Step by step tutorial including several examples of the well-ordering principle for reference.
Overview
The well-ordering principle states that every nonempty subset of the set of all natural numbers has a smallest element. This is possible because the number zero is not included in the set of natural numbers, and therefore cannot appear in a subset of all natural numbers. The well-ordering principle is equivalant to the Principle of Mathematical Induction, but they are proved in different ways and have different sets. Sometimes it is a better idea to use the Well-Ordering Principle, and other times it is a better idea to use the Principle of Mathematical Induction.
Tags: discrete math, element, induction, mathematical, n!, natural, nonempty, number, ordering, PMI, principle, set, smallest, subset, well, well-ordering, WOP
Posted in Discrete Math | No Comments »
Thursday, October 29th, 2009
Order Properties of Natural Numbers
Description
A detailed tutorial on the order properties of natural numbers. Step by step tutorial including several examples of the order properties of natural numbers for reference.
Overview
The order properties are one of the eight sets of properties of natural numbers. The order properties are all based off of inequalities and how to order inequalities. Less than and less than or equal to are the two that are used in the order properties. There are five order properties in all. Since the order properties are of natural numbers, in order to prove the order properties your examples must be natural numbers, or positive integers greater than or equal to one.
Tags: arithmetic, equal, greater than, greater than or equal to, inequalities, less than, less than or equal to, n!, natural, number, order, property, x, y, z
Posted in Arithmetic | No Comments »
Tuesday, October 27th, 2009
The Cross Product of Vectors
Description
A detailed tutorial on the cross product of two vectors. Step by step tutorial including several examples of how to find the cross product for reference.
Overview
A cross product is very similar to a dot product. However, the result of a cross product is a vector, and the result of a dot product is a scalar. In mathematical terms, the cross product can be defined as 
. Theta represents the meausre of the angle between a and b, and n is a unit vector perpendicular to both a and b. The vector this forms is a right-handed system.
Tags: a, algebra, b, cross, dot, n!, outer, perpendicular, product, right-handed, rule, scalar, system, unit, vector
Posted in Algebra | No Comments »
Friday, October 23rd, 2009
The Notation of Basic Number Sets
Description
A detailed tutorial on basic number sets. Step by step tutorial including several examples of the notation of basic number sets for reference.
Overview
There are four basic number sets – N, Z, Q, R. N belongs to Z, and Z and Q belongs to R. This means N also belongs to R. N is the set of all natural numbers. Z is the set of all integers. Q is the set of all rational numbers. R is the set of all real numbers. All the notations of these sets were picked because they relate to certain words. N and R were chosen because they stand for natural and real – which is what the sets are. Q means quotient, because rational numbers are a quotient of any integer provided the denominator is not 0. Z was picked because it stands for zahlen – a German word meaning numbers, and Z is indeed a set of (almost) all numbers.
Tags: all, arithmetic, integer, n!, natural, notation, number, Q, quotient, r, rational, real, set, z, zahlen
Posted in Arithmetic | No Comments »
Thursday, October 8th, 2009
Introduction to the Principle of Mathematical Induction
Description
A detailed tutorial of the principle of mathematical induction. Step by step tutorial including several examples of the principle of mathematical induction for reference.
Overview
The principle of mathematical induction is basically a method of proof-writing, which involves trying to prove that a certain statement is true for all natural numbers. The first statement will be proved, and then the next statement, and the next one. In this way, it is similar to a proof by exhaustion. However, since the statement must be proven for all numbers, eventually an integer will be used in the calculations. This should not be confused with mathematical induction – the principle of mathematical induction is actually a type of deductive reasoning.
Tags: deductive, discrete math, exhaustion, induction, interger, k, Math, mathematical, n!, natural, number, principle, proof, reasoning, statement
Posted in Discrete Math | No Comments »
Thursday, October 1st, 2009
Introduction to Fermat’s Last Theorem
Description
A detailed tutorial of Fermat’s Last Theorem. Step by step tutorial including several examples of Fermat’s Last Theorem for reference.
Overview
Fermat’s Last Theorem is one of the most well known mathematical theorems. Fermat’s Last Theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. Notice that the pattern for this theorem follows the Pythagorean theorem. This theorem had to be proved for odd prime numbers, as Fermat had only left that there was the special instance of n = 4 that works for this equation. Fermat first came up with the problem in 1637, but it was not solved until 1995. This theorem led to the developement of both algebraic number theory and the proof of the modularity theorem.
Tags: a, algebraic number theory, Andrew Wiles, b, c, Calculus, Fermat's Last Theorem, integers, Math, modularity theorem, n!, numbers, odd, Pierre de Fermat, positive, prime, pythagorean theorem
Posted in Calculus | No Comments »
