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Numbers: Rational Numbers

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 Rational numbers as the name suggests are ratios of two numbers. They are derived from Integers and are defined as follows. Let $p,q$ be two integers, i.e. $p,q \in \mathbb{Z}$, then $\frac{p}{q}$ where $q \neq 0$ is a rational number.   in more advanced mathematical notation the rational numbers can be defined as follows $\mathbb{Q}=\{\frac{p}{q}: p,q \in \mathbb{Z}, q \neq  0 \}$                                                           $(1)$ It can be shown that the set of rational numbers is closed with respect to  addition subtraction  multiplication division Let us discuss the closure of rational numbers with respect to the above mentioned operations in detail. The aim here is to show some very basic mathematical proofs. But first a question arises. How do we add two rational numbers? Let $\frac{a}{b},\frac{c}{d}$  be two rational numbers, then we can add them as follows $\frac{ad+bc}{bd} \Longleftrightarrow \frac{a}{b} +\frac{c}{d}$                                              
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Numbers: Integer and their Physical Applications

 Once we have the concept of natural numbers it is very easy to construct negative natural numbers. Mathematically, we just attach a sign "-" to each of the natural numbers except $0$. We define using the rules of addition that  if $a \in  \mathbb{N}$, then $a+(-a)=0$ Combining both the positive and the negative natural numbers we can construct a set which we call the Integers. $\mathbb{Z}=\{\cdots ,-4,-3,-2,-1,0,1,2,3,4, \cdots \}$                                                                     $(1)$ It can easily be shown that this set is closed under  addition  multiplication subtraction Here as discussed earlier, closure of $\mathbb{Z}$ under these operations means that adding, multiplying and subtracting two Integers yields another integer i.e. $+:\mathbb{Z}\times \mathbb{Z} \longrightarrow \mathbb{Z}$ $-:\mathbb{Z}\times \mathbb{Z} \longrightarrow \mathbb{Z}$ $\times:\mathbb{Z}\times \mathbb{Z} \longrightarrow \mathbb{Z}$  The closure of $\mathbb{Z}$ under addition
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Numbers: Mathematical Operations on Natural Numbers

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 In this discussion we will look at the different mathematical operations defined on the set of natural numbers. We denote the set of all natural numbers as follows $\mathbb{N} = \{0,1,2, \cdots \}$                                                                       $(1)$ On this set we define two mathematical operations Addition  Multiplication It must be noted that we are only looking at the operations that are closed .  Let us explain this notion further. Addition as an operation takes two natural numbers and maps them to an other natural number. In mathematical notation we can write as follows. Let $a,b,c \in \mathbb{N}$, where $\in$ means "is an element of" .Then,       $a+b =c$                                                                                                        $(2)$ or operator notation ,        $+(a,b)$                                                                                                         $(3)$ Here $(a,b)$ is called an ordered pai
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Numbers:Natural Numbers

Natural numbers are the most basic form of numbers. These arise from the fact that in our physical universe there exist objects that can be counted. These objects must have the following two properties to enable counting: Distinguishablity : these objects have some sort of identity i.e. they are different from others Persistence : the objects are stable and exist for a sufficient amount of time. Example: Let us take the example of apples in a box. Each apple has a boundary which is separated from the others. Thus we can easily distinguish between two apples. Moreover, each apple is stable and exists for a sufficiently long period of time. An apple will not suddenly disintegrate in front of our eyes, or change form etc. One can see that in a universe where the physical objects did not posses such qualities the concept of counting numbers would not occur naturally. What are these Natural Numbers? These are written as $\mathbb{N}=\{0,1,2, \cdots \}$, where $0$ is a modern inclusion. When
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