Posts Tagged ‘natural’
Tuesday, December 29th, 2009
Overview of the Trichotomy Property
Description
A detailed tutorial on the trichotomy property. Step by step tutorial including several examples of the trichotomy property for reference.
Overview
The trichotomy property is one of the ordering properties of natural numbers. It tells us what order you need to put the natural numbers in – in other words, it tells you the placement of each element of the set of natural numbers. The trichotomy property states that is there are two natural numbers m and n, that m must be either less than n, equal to n, or greater than n. The smaller number is to be placed first, with the larger number after it. If the numbers are equal, then only one number needs to be included as part of the set.
Tags: arithmetic, element, equal, greater, inequality, larger, less, natural, number, order, placement, property, set, smaller, than, trichotomy
Posted in Arithmetic | No Comments »
Friday, December 18th, 2009
How to Define Cardinal Numbers
Description
A detailed tutorial on the definition of cardinal numbers. Step by step tutorial including several examples of how to define cardinal numbers for reference.
Overview
Cardinal numbers are natural numbers that are used to measure cardinality of sets. Cardinality is a fancy way of saying the size of a set. This means the cardinality is the number of elements in a set, provided that the set is finite. If the set is infinite, something called a transfinite cardinal number is used to describe the cardinality of the set. Cardinal numbers are a very important part of set theory, even though they are not studied often or used constantly.
Tags: abstract, algebra, analysis, cardinal, cardinality, combinatorics, elements, finite, infinite, mathematical, measure, natural, number, set, set theory, size, transfinite
Posted in Algebra | No Comments »
Thursday, November 12th, 2009
How to Identify Perfect Numbers
Description
A detailed tutorial on how to identify perfect numbers. Step by step tutorial including several examples of perfect numbers for reference.
Overview
A perfect number is a number that is the sum of all it’s divisors (excluding the number itself, which is also a proper divisor). The way that you identify a perfect number is to find all of its divisors. Once you have them all, add them together. If they equal the number, then it is a perfect number. If they don’t, then it is not a perfect number.
Tags: add, addition, arithmetic, division, divisor, excluding, identify, integer, natural, number, perfect, proper, real, sum
Posted in Arithmetic | No Comments »
Thursday, November 5th, 2009
Introduction to Nested Intervals
Description
A detailed tutorial on nested intervals and the nested interval theorem. Step by step tutorial including several examples of nested intervals for reference.
Overview
Nested intervals means to have one interval (or multiple intervals) inside of another interval. The intervals will get smaller and smaller the more you add, until they will finally dimish entirely. There is a theorem for nested intervals, called the nested interval theorem. It states that if A_n = [a_n, b_n] is a sequence of closed intervals such that A_n+1 is a subset of A_n for all n belonging to the set of natural numbers, then the union over A_n is not an empty set.
Tags: algebra, closed, empty, interval, natural, nested, number, open, sequence, set, subset, theorem
Posted in Algebra | No Comments »
Thursday, November 5th, 2009
Cancellation Properties of Natural Numbers
Description
A detailed tutorial on cancellation properties of natural numbers. Step by step tutorial including several examples of cancellation properties for reference.
Overview
Cancellation properties of natural numbers state that when two terms are equal to each other, if the same number is being multiplied or added on both terms, you may cancel them out and the terms will still be equal to each other. Knowledge of the cancellation properties is extremely important for simplification of equations and when trying to find the value of a variable. Mathematically stated, the cancellation properties are that if x + z = y + z or xz = yz, then x = y.
Tags: add, arithmetic, cancel, cancellation, equal, multiply, natural, number, out, properties, property, simplification, simplify, term, value, variable
Posted in Arithmetic | 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
Successor Properties of Natural Numbers
Description
A detailed tutorial on the successor properties of natural numbers. Step by step tutorial including several examples of the successor properties of natural numbers for reference.
Overview
The successor properties are one of eight sets of properties of natural numbers. The successor properties deal with the actual set of natural numbers, not just parts of the set. It especially concerns the placement of the number 1 in the set of natural numbers. As the term successor implied, these properties deal with what numbers are successors of other numbers. They can be proven by the definition of a successor and the set of natural numbers.
Tags: -1, after, arithmetic, follows, natural, number, properties, set, successor, unique, x
Posted in Arithmetic | 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 »
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 22nd, 2009
Inductive Sets in Set Theory
Description
A detailed tutorial on inductive sets in set theory. Step by step tutorial including several examples of inductive sets in set theory for reference.
Overview
An inductive set is a continuous set of natural numbers that follows a basic pattern of n + 1. This means that for all numbers in the set, that number plus the number one must also be included in the set.The set does not need to include all natural numbers – that is, the set may start at any natural number provided it is greater than or equal to one. However, the set must continue to infinity or it cannot be considered an inductive set.
Tags: -1, addition, complete, continuous, discrete math, element, equal, greater, induction, inductive, infinity, mathematical, natural, numbers, one, pattern, principle, set, subset, theory
Posted in Discrete Math | No Comments »
