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Real numbers are the union of rational numbers and irrational numbers. They contain all positive and negative numbers. Real numbers are denoted by R.

Real numbers are all the numbers whether they are natural, decimal, fractional, rational or irrational and also all positive and negative. You must have heard about Complex numbers, so these complex numbers are the imaginary numbers. In this article we are going to study about the real numbers and methods for solving these problems.

Characteristics of Real Numbers

  1. Real Numbers can be positive or negative.
  2. The sum and product of two real positive numbers will give positive real numbers.
  3. Above two are basic properties of real number. The following are the main properties of real numbers.
  1. Commutative Property
  2. Associative Property
  3. Distributive Property
  4. Identity Property
  1. Commutative Property : According to it, if a and b belongs to real number then, a + b = b + a (for addition )

               a . b = b . a (for multiplication)

  1. Distributive Property : This property is for three real numbers, if a , b and c are belong to real number then, 

a ( b + c) = ab + b c

(a + b) c = ac + b c

  1. Associative Property : This property is for three numbers, if a, b and c belongs to real numbers then,

a + (b + c ) = ( a + b) + c

(ab)c = a (bc)

  1. Identity Property: There are additive and multiplicative identities. for real numbers. If a is any real number
  1. For Addition :  a + 0 = a. (0 is the additive identity)
  2. For Multiplication : a × 1 = 1 × a ( 1 is the multiplicative identity for real numbers)

How to Solve Real Number Problems

We can understand methods for solving real number problems by taking the given example.

Real number problems are of different types, we have taken one type of problem below:-

  1. You have two irrational numbers 3 and 5 . Find the sum of these two irrational numbers, is it rational or irrational.

Sol:  Step – 1: Check whether given numbers are rational or irrational.

  Step – 2: For irrational numbers, sum and subtraction of irrational numbers are also irrational.

Hence, Sum = 3 + 5

  1. Find three rational numbers between 1/3 and 1/2.

Sol: Step – 1: Write the two numbers with any variables like the following.

 Step – 2: Make the denominator of both given numbers the same. You can do it by taking LCM of two numbers which are in denominator.

B = ½ , A = 1/3

   LCM of 2 and 3 is 6.

Step – 3: In order to equalize the denominator of both numbers, multiply the numerator and denominator such that the denominator will get equal to LCM of two numbers.

A = 1/3 × 2/2 = 2/6

B = ½ × 3/3 = 3/6

Step – 4: Now multiply A and B (given numbers after equalizing denominator) by n+1 (n is the quantity of numbers that we have to find between given numbers).

Here, n is 3.

A = 2/6 × 4 = 8/6

B = 3/6 × 4 = 12/6

Step – 5: In this way, you will get the three rational numbers between 1/3 and ½ which are written below.

 Numbers = 9/6 , 10/6, 11/6.

  1. You have two irrational numbers 2 and 3.  Is the product of these two irrational numbers rational or irrational ?

Sol:  I. When you are finding the product or division of both irrational numbers or both rational numbers, then you will get the result as a rational number.

  1. When you have one rational and one irrational number then you will get the result as an irrational number, you can see this example.

a = 2

b = 3

Product = 23

And it is irrational

Division = 2 / 3

It is also an irrational number.

Hence, we can say that the product and division of rational and irrational numbers are irrational numbers.

  1. What should we multiply with 2.25 to get the product equals to 2 ?

Ans: Step-1: For these questions, assume the number to be multiplied x.

Number to be multiplied = x

Step -2:                                    2.25 × x = 2

Step – 3: Isolate the variable terms at one side of equals to and send another terms to another side.

x = 2/2.25

x = 0.88

Step -4: Hence, on multiplying 0.88 with 2.25 we will get product as 2.

Conclusion

In this article we have read about real numbers, their properties and some problems based on them. Real numbers include all natural numbers, whole numbers, decimals and fractional numbers also. Union of irrational and rational numbers forms the whole set of real numbers. There are different types of numbers which are included in the real numbers. They are the rational numbers, integers, whole numbers, etc. We have also gone through the characteristics of real numbers which have been discussed in detail. Here, in this article, we have mainly focused on the step-by-step solution to solve the problems of real numbers which will be easy to understand.