Mathematics is all about numbers. It can be said that Mathematics is nothing without numbers. They are the backbone of Mathematics. From counting to calculation, numbers play a significant role in everyone’s daily life.
Mathematics is one of the major subjects linked with every person from their elementary classes. You are aware of different number systems that express numbers. There are various number systems in Mathematics, such as binary, decimal, etc. That is classified into two sets and then further divided between them. So, finally, you get to know about different types of numbers like natural numbers, integers, whole numbers, real numbers, rational numbers, etc.
In this article, we will be discussing real numbers and some of their related questions.
In Mathematics, various number systems are divided into two categories, i.e., Real numbers and Imaginary numbers. Suppose a number can be represented by the number line. In that case, it is categorised under the Real number section. If you find any number hard to represent on the number line, it will be directed under the Imaginary number section.
Real numbers are just the sum of rational and irrational numbers in the number system. Let’s look at the real numbers more to understand properly.
What Are Real Numbers?
Any number that we can think of, except complex numbers, is a real number. Be it a rational number, fraction, whole number, natural number, integer, or irrational number. For example, real numbers are 3, 0, 1.5, 3/2, √5, etc.
Real numbers can be positive or negative. In Mathematics, it is denoted by the letter ‘R’.
Different types of real numbers are discussed below.
ALSO READ:
NCERT Solutions for Class 10 Maths Chapter 1- Real Numbers
Types Of Real Numbers
Real numbers are classified into different types of number sets. Each number set has a different definition and symbol by which they differ from others.
The table represents the different types of real numbers sets and their examples.
| Type | Definition | Example |
| Natural Numbers (N) | All counting numbers starting from 1 are classified under the natural number set.
N = {1, 2, 3, 4, …} |
2, 34, 456, 5755, …. |
| Whole Numbers (W) | All-natural numbers and 0 are collectively known as whole numbers.
W= {0, 1, 2, 3, …} |
0, 1, 424, 4657, … |
| Integers (Z) | A set including all positive and negative natural numbers with 0 is known as an integer set.
Z = {-∞, …, -1, 0, 1, …, +∞} |
-44, 35, -436, 0, 557, … |
| Rational Numbers (Q) | All numbers that can be written in p/q, where q≠0, are called rational numbers. | 23/4, 565/11, 546/17, 686/7, … |
| Irrational Numbers (P) | An irrational number is defined as a number that cannot be expressed in the form of p/q. | √23, √13, , … |
Also, go through our web story on Real Numbers.
Properties Of Real Numbers
Real numbers show five properties. These are given below:
- Closure Property
- Commutative Property
- Associative Property
- Distributive Property
- Identity Property
Suppose a, b, and c are real numbers. Then the above properties are described as follows:
- Closure Property
The closure property states that the sum and product of two real numbers is always a real number. In simple words, it can be defined as,
If a and b are real numbers then,
a + b = Real number
a * b = Real number
For example:
If a = 12, b = 34
a + b = 12 + 34 = 34 + 12 = 46
a * b = 12 * 34 = 34 * 12 = 408
- Commutative Property
The word commutative is derived from ‘commute’, which means ‘move around’. This property means that if the numbers needing to be operated are changed or swapped from their position, the answer should remain the same.
This property is applicable in the addition of numbers and multiplication of numbers.
- The commutative property for addition:
This property can be defined in simple words as,
a + b = b + a
For example:
24 + 65 = 65 + 24
- The commutative property for multiplication:
In Mathematical form, this property can be defined as,
a * b = b * a
For example:
24 * 65 = 65 * 24
- Associative Property
The word associate itself means ‘to associate’ or ‘ to attach’. This property means that we have three numbers, and they need to either add or multiply. Then it can perform either way or arrangement of numbers.
This property is applicable in the addition of numbers and multiplication of numbers.
- Associative property for addition:
In simple mathematical form, this property is defined as,
a + (b + c) = (a + b) + c
For example:
12 + (34 + 46) = (12 + 34) + 46
- Associative property for multiplication:
In simple mathematical form, this property is defined as,
a * (b * c) = (a * b) * c
For example:
12 * (34 * 46) = (12 * 34) * 46
- Distributive Property
For three real numbers, a, b, c, then the multiplication of real numbers is distributive over addition, is define as,
a * (b + c) = (a * b) + (a * c)
For example:
23 * (12 + 45) = (23 * 12) + (23 * 45)
- Identity Property
Identity property is further divided into two types,
- Additive Identity
If any real number is added with 0 then the answer remains the same real number. In simple words,
a + 0 = 0 + a = a
For example:
23 + 0 = 0 + 23 = 23
- Multiplicative Identity
If any real number is multiplied by 1 then the answer remains the same real number. In simple words,
a * 1 = 1 * a = a
For example:
23 * 1 = 1 * 23 = 23
Sample Question On Real Numbers
Here are some sample problems with real numbers. These samples will help you understand more about the real number.
Some problems are given below:
Example 1: HCF and LCM of the two numbers are 9 and 459, respectively. If one of the numbers is 27, find the other number.
Solution:
Let the second number be ‘x’.
Now, we know that,
HCF * LCM = first number * second number
Putting values in it,
9 * 459 = 27 * x
x = (9 * 459) / 27
x = 4131 / 27
x = 153
Hence, the second number is 153.
Example 2: Express 7/64, 12/125 and 451/13 in decimal form.
Solution:
In 7/64, we have,
Numerator = 7 and Denominator = 64
Now, Dividing 7 by 64, we get,
7/64 = 0.1094
In the similar way, we get,
12/125 = 0.096
451/13 = 34.692
Hence, the decimal forms of 7/64, 12/125 and 451/13 are 0.1094, 0.096, and 34.692.
Example 3: Can two numbers have 15 as their HCF and 175 as their LCM? Give reasons.
Solution:
No, it’s not possible.
LCM = Product of the highest power of each factor involved in the numbers.
HCF = Product of the smallest power of each common factor.
We can conclude that LCM is always a multiple of HCF, i.e., LCM = k × HCF
We are given that,
LCM = 175 and HCF = 15
175 = k × 15
⇒ 11.67 = k
But in this case, LCM ≠ k × HCF
Therefore, two numbers cannot have LCM as 175 and HCF as 15.
Example 4: Prove that 3 + 2√5 is irrational.
Solution:
Let 3 + 2√5 be a rational number.
Then,
The co-primes x and y of the given rational number where (y ≠ 0) is written as,
3 + 2√5 = x/y
Now,
On rearranging them, we get,
2√5 = (x/y) – 3
√5 = ½ * [(x/y) – 3]
Since x and y are integers,
Thus, ½ * [(x/y) – 3] is a rational number.
Therefore,
√5 is also a rational number.
But, this confronts the fact that √5 is irrational.
Thus, our assumption that 3 + 2√5 is a rational number is wrong.
Hence, 3 + 2√5 is an irrational number.
Example 5: Three alarm clocks ring at 4, 12 and 20 minutes. If they start ringing together, after how much time will they next ring together?
Solution:
To find the time when the clocks will next ring together,
We have to find an LCM of 4, 12 and 20 minutes.
4 = 2 x 2
12 = 2 x 2 × 3
20 = 2 x 2 × 5
LCM of 4, 12 and 20 = 22 × 3 × 5 = 60 minutes.
So, the clocks will ring together again after 60 minutes or one hour.
Example 6: Prove that one of every three consecutive positive integers is divisible by 3.
Solution:
Let the three consecutive positive integers be 6, 7, and 8.
Now, according to the question,
We will check whether any one of these consecutive positive integers is divisible by 3 or not.
Therefore,
6/3 = 2
7/3 = 2.333
8/3 = 2.667
Here, we get,
6 is completely divisible by 3.
That means 3 is the multiple of 6.
Hence, we can say that one of every three consecutive positive integers is divisible by 3.
Conclusion
The number system in Mathematics is a vast topic to understand. And real numbers are one of the parts of it. These numbers can perform all the arithmetic operations.
Real numbers are the backbone for understanding number systems and aid in mathematical calculations at all levels of mathematics. The ‘Number System’ topic consists of 6 marks in Class 10 Term 1 board exams.
We tried to explain real numbers, their properties, and applications using properties. We hope that this article will be useful for you.
FAQs
1. How to find three or more rational numbers between any two numbers?
To find three or more rational numbers between any two numbers, we must assume first.
Let two rational numbers be ⅖ and ¾.
And, you have to find three rational numbers in between them.
Take the LCM of denominators of both numbers.
LCM of 5 and 4 = 20
Now,
We will solve the numerator accordingly. After that, we get,
⅖ = 8/20 and ¾ = 15/20
Now,
We have rational numbers lying between them as,
9/20, 10/20, 11/20, 12/20, 13/20, 14/20.
You can write any three from them.
2. What is the difference between real numbers and integers?
The difference between real numbers and integers is as follows:
| Real Numbers | Integers |
| Real numbers include natural, whole, integers, rational, and irrational numbers. | Integers include positive, negative, and zero numbers. |
| The set of real numbers is denoted as ‘R’. | The set of integers is denoted as ‘Z’. |
| Decimals and fractions are also included in real numbers. | Integers do not include decimals and fractions. |
| Examples: ½, , 0.45, etc. | Examples: -2, 45, 0, -12, etc. |
3. How to display a real number on the number line?
To represent rational numbers on the number line, you need to follow the given steps:
Step 1: Draw a horizontal line with arrows on both ends and mark 0 as the origin in the middle of the line.
Step 2: Mark an equal length on both sides of the origin and label it with equal numbers.
Step 3: It should note that the positive numbers lie on the right side of the origin, and the negative numbers lie on the left side of the origin.
Step 4: Mark the numbers on the number line.
4. Name the properties in the following questions:
- 6*(2*3) = (6*2)*3
- 34+ 23 = 23+34
- 4*(3+2) = (4×3) + (4×2)
- To find the property in the given questions,
Let’s compare them with the properties of real numbers.
We know that,
Commutative Property: a + b = b + a, a * b = b * a
Associative Property: a + (b + c) = (a + b) + c, a * (b * c) = (a * b) * c
Distributive Property: a * (b + c) = (a * b) + (a * c)
On comparing, we get,
- 6*(2*3) = (6*2)*3 shows Associative Property
- 34 + 23 = 23 + 34 shows Commutative Property
- 4*(3+2) = (4×3) + (4×2) shows Distributive Property
5. What is the product of non-zero rational and an irrational number?
Suppose the non-zero rational number is 6/5 and the irrational number is .
Now, according to the question,
The product of a non-zero rational number and an irrational number is
6/5 * = (6 * )/5
We know that,
= 3.14
So, (6 * )/5 = (6 * 3.14)/5
= 3.768
And it is a decimal number.
Hence, we can say that the product of a non-zero rational number and an irrational number is always a real number.
