# REAL NUMBERS - DIFFERENT SETS OF NUMBERS FORMING THIS SET, PROPERTIES

Rational Numbers

and

Irrational Numbers before Real Numbers.

We have

Set of Natural Numbers, N
= {1, 2, 3, ......}

Set of Whole Numbers, W
= {0, 1, 2, 3, ......}

Set of Integers, Z
= {... -3, -2, -1, 0, 1, 2, 3, ...}

Set of Rational Numbers, Q
= {x/x = pq where p, qZ, q ≠ 0}

Set of Irrational Numbers, Q'
= { All surds and numbers like π, e etc.
which in decimal form are
non-terminating and non-repeating decimals. }

## Set of all the above Numbers

Set of Real Numbers, R
= The set comprising of all the above Numbers.

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### The Real Numbers set R in chart form

The set R in chart form is shown below.

```

Real Numbers(R)
---------------
⇓             ⇓
Rational Numbers(Q)       Irrational Numbers(Q')
------------------
⇓                 ⇓
Integers(Z)         Fractions
-------------
⇓            ⇓
Whole Numbers(W)   Negative Integers
--------------
⇓            ⇓
Natural Numbers(N)    Zero ('0')

```

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### Properties w.r.t. Multiplication and Addition : Real Numbers

The properties of different sets of Numberswith respect to Addition and Multiplicationare given below in a tabular form.

 Property Set N Set W Set Z Set Q Addition : Closure Yes Yes Yes Yes Commutative Yes Yes Yes Yes Associative Yes Yes Yes Yes Existence of Identity Element No Yes Yes Yes Existence of Inverse Element No No Yes Yes Multiplication : Closure Yes Yes Yes Yes Commutative Yes Yes Yes Yes Associative Yes Yes Yes Yes Existence of Identity Element No Yes Yes Yes Existence of Inverse Element No No No Yes, except for Zero Distributive Law Yes Yes Yes Yes

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