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 can be shown simply by supposing that $a,b$ are two integers. Then, $a+b$ is always going to be an integer. We can check by various substitutions.
$a=3, b=4 \Longrightarrow 3+4=7$
$a=4, b=-2 \Longrightarrow 4+(-2)=2$
$a=1, b=-3 \Longrightarrow 1+(-3)=-2$
$a=-2, b=-4 \Longrightarrow (-2)+(-4)=-6$
$a=-3, b=3 \Longrightarrow (-3)+3=0$
As, we have mentioned earlier to establish a general rule we must show that it is valid for all the possible cases. If we try to list them as we have shown it will require infinite time and space as there are infinite($\infty$) or specifically $\infty \times \infty$ cases that need to be established. Here logical reasoning presents us another great tool known as the proof. It is a sequence of statements each implying the next one that gives an explanation of why a given statement is true. We start with statements that we know to be true, termed as axioms. Assuming that the axioms hold we establish the validity of the statement in question.
Let us try to establish a proof of the the statement that the sum of two integers is an integer.
Proof: We will define the proof as a list of statements
- Axiom: $0 \in \mathbb{Z}$, i.e. $0$ is an integer. This statement is true by definition as shown in equation $1$.
- Axiom: if $x \in \mathbb{Z}$ then $x+1, x-1 \in \mathbb{Z}$, i.e if $x$ is an integer so are $x+1$ and $x-1$. This statement is also true by definition as we see equation $1$. The difference between the adjacent integers is exactly $1$.
- The integers can be divided into three categories namely zero, positive and negative integers.
- It can be easily shown that if $x$ is an integer then $x+0=x$, thus one of the cases is proved.
- It can be seen that a positive integer $y$ can be written as $1+1+ \cdots + 1$, i.e. sum of $y$ number of $1$s. It can be written as $\sum_{i=1}^{i=y}1$. Hence, $x+y = x+\sum_{i=1}^{i=y}1$ . Which is an integer as per successive application of the axiom stated in the second statement of this proof.
- Similarly, it can be seen that a negative integer $-z$ can be written as $(-1)+(-1)+ \cdots +(-1)$ i.e. sum of $z$ number of $(-1)$s. It can also be written as $\sum_{i=1}^{i=z}(-1)$. Hence, $x+z = x+\sum_{i=1}^{i=z}(-1)$ . Which is an integer as per successive application of the axiom stated in the second statement of this proof.
- Hence it is proved that the sum of two integers is an integer. $\blacksquare$
Note: It is something to be appreciated that a finite set of statements can be used to demonstrate the validity of statements, which would otherwise require infinite computations.
The closure of $\mathbb{Z}$ under multiplication can be seen as the result of its closure under addition. If $a,b$ are two integers then $a \times b$ is equivalent to adding $a$ to itself $b$ times
$a \times b = \underbrace{a+a+ \cdots +a}_\text{b times} = \sum_\limits{i=1}^{i=b}a$
or more specifically $2 \times 3 = 2+2+2$.
Since, $\mathbb{Z}$ is closed under addition it can be easily concluded that it is also closed under multiplication.
Similarly the closure under subtraction also follows directly from the closure of $\mathbb{Z}$ under addition. Let $a,b$ be two integers then $a-b$ is equivalent to $a+(-b)$ i.e.
$a-b = a+(-b)$
or more specifically $4-3 = 4+(-3) =1$.
As the set of integers is closed under addition it is also closed under subtraction.
Here one can note that the set of numbers $\{0,1,2,3,\cdots \}$ was closed under addition and multiplication but not under subtraction. A simple example is that $1,3$ are both natural numbers but $1-3 =-2$ is not a natural number. By adding to natural numbers($\mathbb{N}$) the negative natural numbers, we have constructed the set of integers($\mathbb{Z}$) which as a result is also closed under subtraction.
Physical and Practical Applications
In ancient times the negative natural numbers were not considered numbers in the sense of counting numbers as there were no physical objects that were analogous to $-1,-2,$ etc. The importance of these numbers was restricted to financial matters in trade, taxation and lending. If the inflow of money was represented as positive integers then the outflow can be represented as negative integers or vice versa.
Here we are looking at direct physical relevance of these numbers. We are ignoring the uses such as denoting direction, i.e. if a distance is $a$ in one direction, then it is $-a$ in the opposite direction. Such physical relevance came to light in the last couple of hundred years, with the formulation of the concept of electric charge. The electric charge is integer multiple of a particular value, namely the charge of a proton represented as $e_p$. An electron has a charge of $-1\cdot e_p$ and a Neutron has a charge of $0\cdot e_p$.
In modern physics the proton is also theorized to comprise of quarks. If we take this view the charge still remains integer multiple but of a quantity one third($\frac{1}{3}$) of $e_p$.
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