Understanding of Real Numbers

A real number is one that can be discovered in the real world. Numbers can be found all over the place. Natural numbers are used to count things, rational numbers are used to represent fractions, irrational numbers are being used to calculate the square root of a number, and integers are used to measure temperature, among other things. We will study everything about real numbers & their crucial qualities in this session.

What are Real  Numbers:

Except for complicated numbers, every number we can think of is a real number. Rational numbers, such as positively and negatively integers, fractions, or irrational numbers, are all examples of real numbers. The union of set of rational numbers (Q) with the set of irrational numbers (Q) is the set of real numbers, represented by R. 

As a result, the set of real numbers can be written as R = Q q. Natural numbers, whole numbers, integers, rational numbers, & irrational numbers are all examples of real numbers. 

So, which of these numbers isn’t a genuine number? Non-real numbers are those that are neither rational nor irrational, such as -1, 2 + 3i, and -i. 

Types of Real Numbers:

We know that the real numbers are made up of both rational and irrational numbers. As a result, no real number exists that is neither rational or irrational. It simply means that each number we choose from R is either rational or irrational.

Rational Numbers: 

Any number that can be expressed as a fraction p/q is referred to as a rational number. In a fraction, the numerator is denoted as ‘p,’ and the denominator is denoted as ‘q,’ where ‘q’ is not equal to zero. A natural number, a whole number, a decimal, or an integer are all examples of rational numbers. 1/2, -2/3, 0.5, and 0.333, for example, are rational numbers.

Irrational Numbers:

Irrational numbers are a class of real numbers that cannot be represented as a fraction p/q, where p and q are integers as well as the denominator q is not zero (q0.). (pi) is an irrational number, for example. 3.14159265 = 3.14159265 = 3.14159265 = 3.14159265 = 3. The decimal value in this scenario never stops at any point. So, irrational numbers include 2, -7, and so on.

Real Numbers Symbol:

The sign R is used to represent real numbers. The symbols for other types of numerals are listed below.

Natural numbers (N)

W stands for whole numbers.

Integers (Z)

Q – Numbers that are rational.

Irrational numbers are referred to as Q.

Subsets of Real Numbers:

Except for complicated numbers, all numbers are real. As a result, there are five subgroups of real numbers:

Natural numbers are as follows: The set of natural numbers is made up of all positive counting numbers. 1, 2, 3… are the numbers that make up the number N.

Whole numbers: The set containing natural numbers plus 0 is known as the whole numbers set. W = 0, 1, 2, 3, and so on.

Integers: The set of integers includes all positive counting numbers, negative numbers, and zero. Z =…, -3, -2, -1, 0, 1, 2, 3…, Z =…

Numbers that make sense: Rational numbers are those that can be expressed as a fraction p/q, where ‘p’ and ‘q’ are integers and ‘q’ is not equal to zero. Q = -3, 0, -6, 5/6, 3.23 Q = -3, 0, -6, 5/6, 3.23 Q = -3, 0, -6, 5/6

Irrational numbers include numbers that are square roots of positive rational numbers, cube roots of rational numbers, and so on, such as 2. (- 6) = √2, – √6

Real Numbers and Their Properties

The closure property, associative property, commutative property, and distributive property all apply to the set of real numbers, just as they do to the set of natural numbers and integers. The following are some of the most essential qualities of real numbers.

Closure Property: The sum as well as product of 2 real numbers always is a real number, according to the closure property. The following is a description of R’s closure property: If a, b R, a + b R, and ab R are true, then.

Associative Property: The sum or product of any 3 real numbers remains the very same regardless of the order in which the numbers are grouped. The following is the definition of R’s associative property: If a, b, c R, a + (b + c) = (a + b) + c, and a (b x c) = (a x b) c, respectively.

Commutative Property: The commutative property states that the sum and product of two real numbers stay the same even if the order of the numbers is reversed. The following is how R’s commutative property is expressed: If a, b R, a + b = b + a and a b = b a, then a + b = b + a.

Distributive Property: The distributive property is satisfied by real numbers. Multiplication over addition has the distributive property a (b + c) = (a b) + (a c), while multiplication over subtraction has the distributive property a (b – c) = (a x b) – (a x c).

Conclusion:

The number system, often called as the numerical system, is a system for expressing numbers. The number system is divided into two categories: real and imaginary numbers. The sum of rational and irrational numbers is known as real numbers. In general, all arithmetic operations can be performed on these integers. They can also be expressed as numbers on a number line.

Imaginary numbers, on the other hand, are unreal numbers that cannot be expressed on a number line and are commonly used to represent complex numbers.