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# 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, one-dimensional 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. $N=\left\{1,2,3,4,5,\dots \right\}$ All numbers like 1, 2, 3, 4, 10, 105, etc. Whole Numbers Collection of 0 and natural numbers $W=\left\{1,2,3,4,5,\dots \right\}$ 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 non-terminating and non-repeating in nature, such as $\sqrt{2}$ sqrt(2) or $\pi$ .

## 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
$5+3=8$ ; $3+5=8$
$2\frac{1}{2}+150=152\frac{1}{2};\phantom{\rule{4pt}{0ex}}150+2\frac{1}{2}=152\frac{1}{2}$
$-15+45=30;\phantom{\rule{4pt}{0ex}}45+\left(-15\right)=30$
The commutative property of multiplication is represented as follows:
$a×b=b×a$ , where a and b are any two real numbers that are not zero
Example 2
$3×8=24;\phantom{\rule{4pt}{0ex}}8×3=24$
$\frac{1}{2}×100=50;\phantom{\rule{4pt}{0ex}}100×\frac{1}{2}=50$
$-10×12=-120;\phantom{\rule{4pt}{0ex}}12×\left(-10\right)=-120$

## 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.
$\left(x+y\right)+z=x+\left(y+z\right)$
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:
$p\ast \left(q\ast r\right)=\left(q\ast r\right)\ast r$
Example 4
$\left(5×4\right)×3=20×3=60$
$5×\left(4×3\right)=5×12=60$

## 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
$2\left(4+3\right)=2×4+2×3=8+6=14$
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
$8×1=8$
$3184×1=3184$
$-486×1=-486$
$\frac{2}{13}×1=\frac{2}{13}$

## Get help learning about real numbers

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