CLASSIFICATION OF NUMBERS | SCIENCE TUTOR

Natural numbers

The most familiar numbers are the natural numbers or counting numbers: 1, 2, 3, and so on. There is no universal agreement about whether to include zero in the set of natural numbers. Today some textbooks, especially college textbooks, define the natural numbers to be the positive integers {1, 2, 3, ...}, while others, especially primary and secondary textbooks, define the term as the non-negative integers {0, 1, 2, 3, ...}.

N0=N0={0,1,2,3,...}N*=N+=N1=N>0={1,2,3,...}
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. In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right.

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Integers

An integer (from the Latin integer meaning "whole"), commonly known as a "whole number", is a number that can be written without a fractional component. The set of integers consists of zero (0), the natural numbers (1, 2, 3, ...) and their inverse (negatives, i.e. −1, −2, −3, ...).

Z={...−4,−3,−2,−1,0,1,2,3,4...}
Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, 0, Z (unlike the natural numbers) is also closed under subtraction.

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Rational numbers

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number.

Q=

a

b

wherea,b∈Zandb≠0
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.

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Real numbers

In mathematics, a real number is a value that represents a quantity along a continuous line. 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 two, an irrational algebraic number) and π (3.14159265…, a transcendental number).

R=thelimitofaconvergentsequenceofrationalnumbers
Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one.

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