Some types of numbers(Num-1)

The Natural numbers
The Integers
The Rational numbers
The Real numbers

The Natural numbers

The natural numbers are the oldest kind of numbers used for counting purposes. The modern representation of numbers using $0-9$ called the Hindu-Arabic numerals by west which is wrong as it has nothing to do with Arabs, they only passed on what they had learnt about this representation from Hindus. So they should be called the Hindu numerals. The west always seems to have problem with eastern Hindu-Chinese contribution to mathematics as they don’t recognize it often, for more about it some other day. The set of natural numbers is denoted by $\mathbb{N}$ , it consists of the positive whole numbers (=numbers $>= 0$ without decimals). Some authors prefer to include $0$ to this set. We would not be doing so. So $\mathbb{N}=\{1,2,...,n,...\}$ . This $\mathbb{N}$ forms a semigroup, a set with associative binary operation. The binary operation in this case being usual addition. The semigroups are just abstract version of properties of $\mathbb{N}$ so that they become available in wide array of situations. We sometimes denote $\mathbb{W}=\mathbb{N} \cup \{0\}$ as the set of whole numbers. This forms a monoid which is a semigroup with identity in this case $0$ being additive identity meaning any number added to zero is again the same number.

The Integers

These are the numbers which are positive, negative including zero without fractional part. It is denoted by $\mathbb{Z}$ , from the German word Zahlen. $\mathbb{Z}= \{...,-2,-1,0,1,2,...\}=\mathbb{N}\cup \{0\}\cup -\mathbb{N}$ . They form a group meaning they have a well-defined addition, additive identity which is zero and additive inverses for example additive inverse of $2$ is $-2$ when they are added they give us the additive identity zero. There is one more thing we can do with the integers, the same could be done with the natural numbers too we could multiply them and their multiplication will again be the integer or the natural number. This is called as the integers are closed under multiplication. The integers are example of one more mathematical structure called ring, where addition, subtraction and multiplication makes sense. The need for negative numbers arose from solving quadratic equations. Frankly speaking there is nothing natural about natural numbers or any other numbers they are all just human creation to understand the nature better. They are useful to us in many ways. They help us quantify and classify things at deeper level than humans can understand without it. For example today computers can find patterns in far superior way than humans using mathematical and statistical techniques.

The Rational numbers

The set of rational numbers is basically the ratio of the two integers whenever they make sense, i.e. when denominator is nonzero. It is denoted by $\mathbb{Q}$ , for Quotient. More precisely it is defined as, $\mathbb{Q}=\{\frac{m}{n}: m,n \in \mathbb{Z}, n \neq 0\}$ . These are the numbers closed under addition, subtraction, multiplication and division i.e. when you divide a rational number by another rational number you get another rational number which is not the case for the integers. For example when you divide 2 by 4 you do not get an integer but a rational number 1/2. The numbers satisfying such division for all numbers except for zero are part of what is called a Field. The rational numbers have finite decimal expansion or they have infinite decimal expansion with finite part repeating for infinitely many times, for example $1.123412341234....$ is a rational number because suppose $x= 1.12341234...$ . Then $10^4 x=11234.12341234....$ , now $10^4 x - x=11233$ , so $x = \frac{11233}{9999}$ which is a rational number by definition. The interesting property of the rational numbers is that if you pick any two rational numbers however close they are, there are infinite number of rational numbers between them. This is referred to as the Archimedean property The one important thing that is happening in the background is that we are using usual absolute value for defining these numbers, there are other non-trivial ways to define absolute values on these numbers and they can lead to different number system such as p-adic numbers. We would talk about them in future blogs.

The Real numbers

The set of real numbers is the union of the set of rational numbers and the numbers which have non-recurring infinite decimal expansion. Let me show that using mathematical proof that square root of 2 is not a rational number. Suppose $\sqrt{2}$ is a rational number. Then we could write $\sqrt{2}=\frac{a}{b}$ for some $a,b \in \mathbb{Q}, b\neq 0$ , where $\frac{a}{b}$ is in most reduced form i.e. gcd of $a,b$ is 1. Now $2= \frac{a^2}{b^2}$ , giving $2b^2=a^2$ which means $a^2$ is an even number but we know that square is even implies the number itself must be even because square of an odd number is always odd. So we have $a=2m$ , for some $m \in \mathbb{Z}$ , which means $a^2=4m^2=2b^2 \implies b^2 = 2m^2$ , Therefore $b$ is even as well by similar reasoning as above. But that is contradictory to our assumption that gcd of $a,b$ is 1, which in turn means $\sqrt{2}$ is not a rational number. The set of real numbers is denoted by $\mathbb{R}$ . More precisely we say that the $\mathbb{R}$ is the completion of the $\mathbb{Q}$ with respect to usual absolute value, the completion here means in the sense of the limit or analytical terms not in algebraic terms. It basically fills all the gaps in the rational numbers and makes it the real line without gaps. As can be easily seen that $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ , in particular we call the real numbers which are not rational numbers as irrational numbers. They are denoted by set theoretic notation $\mathbb{R} \backslash \mathbb{Q}$ . The numbers $\pi$ and $e$ are two of the most popular irrational numbers, also square root of any prime number is also an irrational number.

There are two more important set of numbers, one is the set of complex numbers and the other is the p-adic numbers. They need a blog of their own separately. So we would be discussing them in future.

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