Real Numbers
Real numbers are a crucial part of the number system that we use to
represent and perform arithmetic operations on quantities in the
real world. They include all
rational numbers, which are numbers that can be expressed as the ratio of two
integers, and
irrational numbers, which cannot be expressed as the ratio of two integers and have
an infinite number of decimal places. In general, all arithmetic
operations (such as
addition,
subtraction,
multiplication, and
division) can be performed on real numbers.
Real numbers definition
The real numbers are also thought of as the "usual" numbers. Just
think of them as all the union between the set of rational numbers
and the set of irrational numbers. Real numbers can be used to
measure a continuous, onedimensional quantity, such as temperature,
duration, or distance. By continuous, we mean that pairs of values
can have arbitrarily small differences. Every real number can be
almost uniquely represented by an infinite decimal expansion.
There are infinitely many real numbers, just as there are infinitely
many rational or irrational numbers, but it can be proved that the
infinity of real numbers is a bigger infinity.
This Venn diagram shows the relationships between various sets of
numbers, including
whole numbers,
integers,
rational numbers, and irrational numbers, all of which fall into the
category of real numbers.
Set of real numbers
The set of real numbers includes various categories, such as
natural and whole numbers, integers, and rational and irrational numbers.
In the table below, all the real number formulas, or the
representation of the classification of real numbers, are defined
with examples.
Category 
Definition 
Example 
Natural Numbers 
Contains all counting numbers starting from 1.
$N=\{1,2,3,4,5,\dots \}$

All numbers like 1, 2, 3, 4, 10, 105, etc. 
Whole Numbers 
Collection of 0 and natural numbers
$W=\{1,2,3,4,5,\dots \}$

All numbers including 0 such as 0, 1, 2, 3, 4, 5… 
Integers 
The collective result of whole numbers and the negative
of all natural numbers

…4, 3. 2, 1, 0, 1, 2, 3, 4, 5… 
Rational Numbers 
Numbers that can be written in the form of p/q where
q≠0, and p and q are both integers

Examples of rational numbers are 1/2, 6/5, 15/212, etc.

Irrational Numbers 
The numbers that are not rational and cannot be written
in the form of p/q. where p and q are both integers

Irrational numbers are nonterminating and nonrepeating
in nature, such as
$\sqrt{2}$
sqrt(2)
or
$\pi $
.

Properties of real numbers
Real numbers and the commutative property
The commutative property states that the numbers on which we operate
can be moved or swapped from their original position without
changing the resulting answer. This property holds true for addition
and multiplication of real numbers, but not for subtraction or
division.
The commutative property of addition is represented as follows:
$a+b=b+a$
, where a and b are any two real numbers
Example 1
$5+3=8$
;
$3+5=8$
$2\frac{1}{2}+150=152\frac{1}{2};\phantom{\rule{4pt}{0ex}}150+2\frac{1}{2}=152\frac{1}{2}$
$15+45=30;\phantom{\rule{4pt}{0ex}}45+(15)=30$
The commutative property of multiplication is represented as
follows:
$a\times b=b\times a$
, where a and b are any two real numbers that are not zero
Example 2
$3\times 8=24;\phantom{\rule{4pt}{0ex}}8\times 3=24$
$\frac{1}{2}\times 100=50;\phantom{\rule{4pt}{0ex}}100\times \frac{1}{2}=50$
$10\times 12=120;\phantom{\rule{4pt}{0ex}}12\times \left(10\right)=120$
Real numbers and the associative property
With the associative property, when more than two real numbers are
added or multiplied, the result remains the same, no matter how the
numbers are grouped. The associative property does not hold true for
division or subtraction equations.
The associative property of addition:
$\left(x+y\right)+z=x+\left(y+z\right)$
Example 3
$\left(3+5\right)+16=8+16=24$
$\left(3+5\right)+16=3+21=24$
$\left(4+12\right)+6=8+6=14$
$4+\left(12+6\right)=4+18=14$
The associative property of multiplication:
$p\ast (q\ast r)=(q\ast r)\ast r$
Example 4
$\left(5\times 4\right)\times 3=20\times 3=60$
$5\times \left(4\times 3\right)=5\times 12=60$
Real numbers and the distributive property
The distributive property explains that operations performed on
numbers available in brackets can be distributed for each number
outside the bracket. It is one of the most frequently used
properties in mathematics. When a factor is multiplied by the sum or
addition of two terms, it is essential to multiply each of the two
numbers by the factor, and finally perform the addition operation.
The property is expressed as:
$a\left(b+c\right)=ab+ac$
, where a, b, and c are three different values
Example 5
$2\left(4+3\right)=2\times 4+2\times 3=8+6=14$
You can check your work to make sure that this is correct by solving
the problem as it is originally written:
$2\left(4+3\right)=2\left(7\right)=14$
Since you get the same solution, you can see that this property
works correctly.
Real numbers and the identity property
There is an additive identity and a multiplicative identity as shown
below.
In addition, 0 is the additive identity. The property states that
any real number plus 0 is the number itself.
$m+0=m$
For multiplication, 1 is the multiplicative identity. The property
states that any real number multiplied by 1 is the number itself.
$m\ast 0=m$
Example 6
$46+0=46$
$8347984+0=8347984$
$19+0=19$
$\frac{1}{5}+0=\frac{1}{5}$
Example 7
$8\times 1=8$
$3184\times 1=3184$
$486\times 1=486$
$\frac{2}{13}\times 1=\frac{2}{13}$
Topics related to the Real Numbers
Flashcards covering the Real Numbers
Practice tests covering the Real Numbers
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