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Classification of Numbers

Following are the classifications of numbers.

1. Natural Numbers:

  1. Each of 1,2,3,4,…..,etc is a natural number.
  2. The smallest natural number is 1 ;whereas the largest natural number cannot be obtained.
  3. Consecutive natural numbers differ by 1.
  4. Let x be any natural number , then the natural numbers that come just after x are x+1, x+2, x+3, etc.

2. Even Natural Numbers:

A system of natural numbers ,which are divisible by 2 or are multiples of 2, is called a set of even numbers.

E= (2,4,6,8,10,12……..)

There are infinite even numbers.

3. Odd Natural Numbers:

A system of natural numbers ,which are not divisible by 2 ,is called a set of odd numbers.

O= (1,3,5,7,9………)

There are infinite odd numbers.

Taking together the odd and even numbers, we get natural numbers.

4. Whole Numbers:

  1. 0,1 ,2,3,4,…… etc are whole numbers.
  2. The smallest whole number is zero whereas the largest whole number cannot be obtained.
  3. Consecutive whole numbers differ by 1.
  4. Except zero every whole number is a natural number and because of this:
    1. Every even natural number is an even whole number
    2. Every odd natural number is an odd whole number.

5. Prime Numbers:

  1. Whole numbers greater than 1 that are divisible by unity and itself only.
  2. Except 2 all other prime numbers are odd. P= 2,3,5,7,11,13,………. etc.

6. Composite Numbers:

A composite number is a whole number (greater than 1) that is not prime.

Composite numbers C= (4,6,8,9 ……..,etc)

7. Integers:

  1. The integers consists of natural numbers , zero and negative of natural numbers. Thus , Z or I = …………………,-4,-3,-2,-1, 0 , 1,2,3,4…………….
  2. There are infinite integers towards positive side and infinite integers towards negative side .
  3. Positive integers are the natural numbers.

Use of Integers

The integers are used to express our day-to-day situations in mathematical terms.

  1. If profits are represented by positive integers then losses by negative integers.
  2. If heights above sea level by positive integers then depths below sea level by negative integers.
  3. If rise in price is represented by positive integers ,then fall in price by negative integers and so on.

8. Rational Numbers:

Any number which can be expressed in the form of \dfrac{a}{b} , where a and b both are integers and b \neq 0 , is a rational number.

  1. \dfrac{2}{5} is a rational number, since 2, 5 are integers and 5 is not equal to zero.
  2. \sqrt{2}, \sqrt{3}, \sqrt{5} , etc are not rational numbers since these numbers cannot be expressed as \dfrac{a}{b} .

So, we can say that rational numbers contain all integers and all fractions (including decimals). There are infinite number of rational numbers.

  1. Every integer is a rational number but the converse is not true. The same result is true for natural numbers, whole numbers, fractions, etc.
  2. \dfrac{a}{-b}= \dfrac{-a}{b}=-(\dfrac{a}{b})

9. Irrational Numbers:

Then numbers which are not rational are called irrational numbers .

Each of 3\sqrt{4}, \sqrt{5} , etc is an irrational number.

The number \dfrac{a}{b} is neither rational nor irrational if b=0 .

10. Real Numbers:

Every number, which is either rational or irrational is called a real number.

  1. Each natural number is a real number.
  2. Each whole number is a real number.
  3. Each integer is a real number.
  4. Each rational number is a real number.
  5. Each irrational number is a real number , etc.

Absolute Value of a Number:

The absolute value of an integer is its numerical value regardless of it’s sign.

Absolute value of (-68) =\lvert -68\rvert\ =68

Absolute value of (+47) =\lvert +47\rvert\ =47

Therefore if a represents an integer, its absolute value is represented by \lvert a\rvert\ and is always non-negative

Remember:

  1. \lvert a\rvert\ = a , when a is positive or zero
  2. \lvert a\rvert\ = (-a) , when a is negative.


Comments

    • sumayyaknwl123@gmail.com says

      an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations.

      Reply

  14. Jean-Yves BOULAY says

    According to a new mathematical definition, whole numbers are divided into two sets, one of which is the merger of the sequence of prime numbers and numbers zero and one. Three other definitions, deduced from this first, subdivide the set of whole numbers into four classes of numbers with own and unique arithmetic properties. The geometric distribution of these different types of whole numbers, in various closed matrices, is organized into exact value ratios to 3/2 or 1/1.

    Reply

  15. Jean-Yves BOULAY says

    The ultimate numbers
    Definition of an ultimate number
    Considering the set of whole numbers, these are organized into two sets: ultimate numbers and non-ultimate numbers.
    Ultimate numbers definition:
    An ultimate number not admits any non-trivial divisor (whole number) being less than it.
    Non-ultimate numbers definition:
    A non-ultimate number admits at least one non-trivial divisor (whole number) being less than it.
    Other definitions
    Let n be a whole number (belonging to ℕ), this one is ultimate if no divisor (whole number) lower than its value and other than 1 divides it.
    Let n be a natural whole number (belonging to ℕ), this one is non-ultimate if at least one divisor (whole number) lower than its value and other than 1 divides it.

    Reply

  19. Kashob says

    Thank you so much bro, You explain it like very easy…… I understand all types of numbers easily… I repeat again very very thank you….

    Reply

  21. Jean-Yves BOULAY says

    In your whole numbers ranking you describe primes and composites but you don’t rank numbers 0 and 1 which are whole numbers!

    Here an innovative classification of whole numbers with 0 and 1 clearly classified:

    https://www.researchgate.net/publication/341787835_New_Whole_Numbers_Classification

    According to new mathematical definitions, the set (ℕ) of whole numbers is subdivided into four subsets (classes of numbers), one of which is the fusion of the sequence of prime numbers and numbers zero and one. This subset, at the first level of complexity, is called the set of ultimate numbers. Three other subsets, of progressive level of complexity, are defined since the initial definition isolating the ultimate numbers and the non-ultimate numbers inside the set ℕ. The interactivity of these four classes of whole numbers generates singular arithmetic arrangements in their initial distribution, including exact 3/2 or 1/1 value ratios.

    Reply

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