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A detailed tutorial on the definition of cardinal numbers. Step by step tutorial including several examples of how to define cardinal numbers for reference.

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.

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A detailed tutorial on the pigeon-hole principle. Step by step tutorial including several examples of the pigeon-hole principle for reference.

The pigeon-hole principle is an important principle in math that states that if n items are to be put into m pigeon-holes, and n > m, then at least one pigeon-hole must contain more than one item. It is thought of as an extension of the counting principle. The pigeon-hole principle was first referred to as the drawer principle, or the shelf principle. Because of this, it is commonly called Dirichlet’s box principle or Dirichlet’s drawer principle. It is most commonly used with finite sets of elements; however, this principle can also be used with infinite sets.

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A detailed tale of the Infinite Hotel. Step by step story including several pictures and an explanation of the Infinite Hotel for reference.

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.

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A detailed tutorial on saddle-point approximation. Step by step tutorial including several examples of saddle-point approximation for reference.

Saddle-point approximation is also referred to as the method of steepest descent and Laplace’s method. It is a way of approximating integrals in the form . f(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b have a possibilty of being infinite.

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A detailed tutorial on infinite sets. Step by step tutorial including several examples of infinite sets and how to identify them for reference.

There are two types of sets, finite sets and infinite sets. The tutorial will focus on infinite sets. An infinite set is a set that has at least one endpoint of infinity, which can be implied either by having infinity in the set or by having a trailing end of the set, with no number at the end. Infinite sets can either be countable or uncountable – meaning they either have a pattern you can use to follow to infinity, or there is no pattern present.

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A detailed tutorial on the definition of a finite set. Step by step tutorial including several examples of finite sets for reference.

There are many different types of sets, but one of the most common ones to run into a finite sets. A finite set is a set that has a finite number of elements – meaning a set with a definite number of elements, such as five, or ten. The number of elements in the set must be a natural number, and it is called the cardinality of a set. An empty set is considered to be finite, with a cardiality of zero, even though zero is not considered to be a natural number.

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A detailed tutorial of how to use Pascal’s Triangle. Step by step tutorial including several examples of how to use Pascal’s Triangle for reference. Knowledge of Pascal’s Triangle will prove useful in several branches of mathematics.

Pascal’s Triangle is a useful device in mathematics that can reveal the sums of almost any number. There are an infinite number of rows but it can be shortened to any number. The triangle traditionally starts at Row 0 with one number – 1. Then Row 1 has two numbers – 1 and 1. And Row 2 has three numbers – 1, 2, and 1. The number 1 lines the side of the triangle. Every other number is the sum of the two numbers found directly above it. Pascal’s triangle is constructed by using geometric shapes.