Real numbers are a collection of numbers that includes both rational and irrational numbers. The real numbers on the number line are “all the numbers.” There are an infinite number of real numbers, just as there are infinite numbers in each of the other sets. A collection of real numbers comprises these various types of numbers. We will learn about real numbers and their important properties in this lesson.

Real Number System

Rational numbers, such as positive and negative integers, fractions, and irrational numbers, are all examples of real numbers. Real numbers include things like 3, 0, 1.5, 3/2, 5, and so on.

Forms of Real Numbers

Real Numbers Come in a Variety of Forms. We know that real numbers are made up of both rational and irrational numbers. As a result, no real number exists that is neither rational nor irrational. It simply means that any number we choose from R is rational or irrational.

  • Rational Numbers: Any number expressed as a fraction p/q is rational. In a fraction, the numerator is “p,” and the denominator is “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. For example, rational numbers include 1/2, -2/3, 0.5, and 0.333.
  • Irrational Numbers:  A class of real numbers cannot be expressed as a fraction p/q, where p and q are integers, and the denominator q is not zero (q≠0.). (pi) is an irrational number. The decimal value, in this case, never ends at any point. As a result, irrational numbers include 2, -7, and so on.

Real Numbers Symbol

The symbol R is used to represent real numbers. The symbols for the other types of numbers are listed below: 

  • Natural numbers (N)
  • W stands for whole numbers.
  • Integers (Z)
  • Q – Rational Numbers.
  • Irrational numbers – P

Real Numbers Properties

The closure, associative, commutative, and distributive properties 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 important properties of real numbers.

  • Closure Property: According to the closure property, the sum and product of two real numbers are always real. The following description of R’s closure property: If a, b belongs to R, a + b belongs to R, and ab belongs to R are true.
  • Associative Property: The sum or product of any three real numbers remains the same regardless of the order in which they are grouped. The following is the definition of R’s associative property: If a, b, and c are R, a + (b + c) = (a + b) + c, and a (bc) = (a b) + c, respectively. 
  • Commutative Property: states that the sum and product of two real numbers remain 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  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 b) – (a c).

Real number system Diagram

The real numbers and their categorisation can be easily understood by several real number system diagrams available online to give you an easy understanding.

Real Number System 

Real Number System : Each real number belongs to one or more sets below. You should be able to recognise the numbers described in the table. Knowing the names of these numbers can help describe domains of functions and comprehend theorems like the rational zeros theorem. You can use various sites, and real number system notes Pdfs to learn real number systems.

Conclusion

Real numbers are decimals, with an infinite number of decimal places used to measure continuous quantities. The second point is that any real number that satisfies the axioms is an upper bound, whereas rational numbers are not. The third point is that all Cauchy sequences converge on real numbers. Real numbers are a collection of numbers that include both rational and irrational numbers. The real numbers on the number line are “all the numbers.” There are an infinite number of real numbers, just as there are infinite numbers in each of the other sets. You can get several rational number charts online for better and easier understanding.