Number Systems

Number: A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4 and so forth.

Main classification:

The major categories of numbers are as follows:

Main number systems

${\displaystyle \mathbb {N} }$	Natural	0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ... ${\displaystyle \mathbb {N} _{0}}$ or ${\displaystyle \mathbb {N} _{1}}$ are sometimes used.
${\displaystyle \mathbb {Z} }$	Integer	..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
${\displaystyle \mathbb {Q} }$	Rational	a/b where a and b are integers and b is not 0
${\displaystyle \mathbb {R} }$	Real	The limit of a convergent sequence of rational numbers
${\displaystyle \mathbb {C} }$	Complex	a + bi where a and b are real numbers and i is a formal square root of −1

Natural numbers:

The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. 

At present, In the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The radix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 10).

Integers:

An integer (from the Latin integer meaning "whole") is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, ​5 1⁄2, and √2 are not.

Rational number:

A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. 

Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator.

The fraction m/n represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is:

${\displaystyle {1 \over 2}={2 \over 4}.}$

In general,

${\displaystyle {a \over b}={c \over d}}$ if and only if ${\displaystyle {a\times d}={c\times b}.}$

Real numbers:
A real number is a value that represents a quantity along a line.The real numbers include all the measuring numbers. The symbol for the real numbers is R.

The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number).

Complex numbers:

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x² = −1, which is called an imaginary number because there is no real number that satisfies this equation. For the complex number a + bi, a is called the real part, and b is called the imaginary part.

Subclasses of the integers:

Even and odd numbers:

An even number is an integer that is "evenly divisible" by two, that is divisible by two without remainder; an odd number is an integer that is not even

Prime numbers:
A prime number is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11