Illustrates the different subsets of real numbers

Illustrates the different subsets of real numbers. Rational numbers set (Q) and irrational numbers set (Qˊ) are the main subsets of the Real numbers.

Illustrates the different subsets of real numbers

Real numbers have two subsets. The rational numbers set is a subset of the Real numbers set. The irrational numbers set is another subset of the Real numbers set. The whole numbers set is the subset of the Rational numbers set. The natural numbers set is a subset of the Whole numbers set.

Different subsets of real numbers

Natural numbers

1, 2, 3, 4… Numbers are used for counting. These are called Natural numbers. Also known as positive whole numbers or counting numbers. The set of all the whole numbers is represented with N.

  • -1 is not a natural number
  • 0.5 is not a natural number
  • 1 is a natural number
  • 0 is a natural number
  • -0.5 is not a natural number

Whole numbers

Whole numbers are any of the numbers like…...-3, -2, -1, 0, 1, 2, 3,…… etc. Whole numbers will not contain any fractional or decimal points. Set of whole numbers are represented with Z. 

From the discussion of natural numbers, we can say N(Natural numbers) is a subset of Z(Whole numbers). There are unlimited whole numbers.

A decimal or fractional number lies between two whole numbers but remember that fractional is not the whole number.

Example:
  • 5 is a whole number
  • 43549043 is a whole number
  • -5 is a whole number.
  • 0 is a whole number
  • 0.6 is not a whole number.
  • -1.6 is not a whole number.

The set of natural numbers is the subset of the whole number set.

N⊂Z

Rational number

If any number can be represented by p/q  (where p and q both are whole numbers and co prime also q ≠ 0) then the number is called Rational number. Set of all Rational number is represented by Q. If rational number is represented in decimal then that will be Terminating decimal or Recurring decimal number.  If q =1 then we can say all the whole numbers are Rational numbers.

A set of Whole numbers(Z) is a subset of rational numbers set(Q).

Z⊂Q

Irrational numbers definition

If any number cannot be represented by p/q (where p and q both are the whole numbers and co prime also q is not equal to 0) then the number is called an Irrational number. The set of all irrational numbers is represented by Qˊ.

If an irrational number is represented in decimal then that will be a Non-terminating decimal or Non-Recurring decimal number. Example: 0.101001000100001..... is an irrational number.

Difference between rational numbers and Irrational numbers


Rational number
Irrational number

If any number can be represented by p/q (where p and q both are the whole numbers and co prime also q is not equal to 0) then the number is called Rational number.
If any number cannot be represented by p/q (where p and q both are the whole numbers and co prime also q is not equal to 0) then the number is called an Irrational number.
If a rational number is represented in decimal then that will be a Terminating decimal or Recurring decimal number.
If an irrational number is represented in decimal then that will be a Non-terminating decimal or Non-Recurring decimal number.

Set of all Rational number is represented by Q.
The set of all irrational numbers is represented by Qˊ.


Real numbers definition

All the rational numbers and all the irrational numbers are totally called Real numbers. The set of all real numbers is represented by R.

So,

  • Natural numbers set (N) is a subset of Whole numbers set (Z)
  • The whole numbers set (Z) is the subset of Rational numbers set (Q)
  • Rational numbers set (Q) is a subset of Real numbers set (R)
  • Irrational numbers set (Qˊ) is also a subset of the Real numbers set (R)

N⊂Z⊂Q⊂R and Qˊ⊂R

Union of Rational numbers set (Q) and Irrational numbers set (Qˊ) equals to Real numbers set(R)

Q∪Qˊ= R

The intersection of Rational numbers set (Q) and Irrational numbers set (Qˊ) equals to empty set()

Q∩Qˊ= ∅

Geometrical discussion

Every straight line is extended on both sides from 0. By the points of these lines, all the real numbers can be represented. This is called the real numbers line.


Real numbers

Follow the figure, O is the main point and if points are represented on both sides of the main point then the points will represent the rational numbers. Each of the points represents a specific rational number.
Let, OPM is an equiangular isotherm ant the two equal arms are one unit in size.

Now, OP = √1² + 1² = √2 = OQ [Same radius]

But √2  is not a rational number. So, like this unlimited number of irrational numbers are on that line.
Overall we can say, in a specific straight line some specific points are the members of the set of rational numbers. The numbers on the left side from O is showing the negative numbers and the right side from O is showing the positive numbers.

Other points will be irrational numbers.

Root 2 is an irrational number



Root 2 is an irrational number


Here are the subsets of real numbers-

subsets of real numbers

Some rules-based on real numbers

  • If a, b are real numbers then a+b is also a real number
  • If a,b,c are real numbers then, (a+b) +c = a+ (b+c)
  • For any real number a another real number b will get if a+b = b+a =0 
  • a+b = b+a
  • If a,b are real numbers then their multiplication will also be a real number.
  • If a,b real numbers then a.b = b.a
  • a.(b+c) = (b+c).a
  • If a and b are real numbers then a>b or a =b or a<b
  • If a,b,c real numbers  a>b and b>c then a>c
  • If a,b,c real numbers and a>b then a+c > b+c
  • If a>b then
  1. ac > bc if c > 0
  2. ac < bc if c < 0

Some properties of the subset of real numbers

Bounded above

If there is a real number M and if it is equal or greater than any elements of real numbers subset S then then then the subset S is called bounded above and M is an upper bound of that set.
Condition: x∈S ⇒M≥x
Example:
If S = { 0,½, 1,2,-1/3, -3/4,-3,3 } then upper bound is 3.
In the set S, multiple numbers of upper bounds can exist. From those which are smaller that is called the least upper bound. It is represented with Sup S.

Bounded below

If there is a real number M and if it is equal or less than any elements of real numbers subset S then then then the subset S is called bounded below and M is a lower bound of that set.

Condition: x∈S ⇒M≤x

Example:

If S = { 0,½, 1,2,-1/3, -3/4,-3,3 } then lower bound is 0.

In the set S, multiple numbers of lower bounds can exist. From those which are bigger that is called the greatest lower bound. It is represented with Inf S.

Bounded set

If a subset from the set of real numbers is both upper-bounded and lower bounded then that is called a bounded set.

Un-bounded set

If a set is not bounded then the set is called an unbounded set.
Remember if a subset of a real number is bounded above then that set has a unique least upper bound and that is a real number.

Absolute value

O is the main point: the distance between A and O and it is called absolute value.

Real numbers

In the linear structure o number, we see,

  • The absolute value of a is = Distance between O and A = a
  • The absolute value of –a is the distance between O and Aˊ = a
  • The absolute value of a is represented with |A|

For all the real numbers

|a|= a, if a>0 or 0 if a = 0 or –a if a<0

Example: If |8|=8 ,|-9|=-(-9) =9

Properties of absolute value

  1. |a|>=0
  2. |ab|= |a|.|b|
  3. |a/b|= |a|/|b|
  4. |a+b|<=|a|+|b|


So, Real numbers are divided like this-

Real numbers

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