Real Numbers

Real Numbers are continuous quantities that can represent a distance along a line, as Real numbers include both rational and irrational numbers. Rational numbers occupy the points at some finite distance and irrational numbers fill the gap between them, making them together to complete the real line. So, in other words, real numbers are those numbers that can be plotted on the real line.

Real numbers include rational numbers including positive and negative integers, fractions, and irrational numbers. Basically, any number that we can think of is a real number.
Examples of Rational Numbers are 2, 3.5, 6/7, √5, 0.35, -67, [Tex]\pi[/Tex], e, etc.

In this article, we will discuss real numbers in detail, including their properties, representation on the number line, and decimal expansion and we will also check if 0 is a real number.

What are Real Numbers?

Real numbers are all the numbers that can be found on the number line. This includes both rational numbers (like 7, -3, 0.5, and 4/3) and irrational numbers (like √2) . They encompass integers, fractions, and decimals, representing a continuous, unbroken set of values.

The collection of all rational numbers contains all the other numbers like natural numbers, integers, rational as well as irrational. Some examples of real numbers are 3 (a whole number), -1 (an integer), 1/2 (a rational number), √2 (an irrational number), πぱい (an irrational number), 2.5 (a decimal number), etc.

The number or the numeral system is the system of representing numbers. There are various types of number systems in maths like binary, decimal, etc. The Number system represents how a number should be written.

Real Numbers Chart

Rational Numbers, Irrational Numbers, and all the subsets of rational numbers all come under real numbers, and the real number chart is used to express all the branches of real numbers. The real number chart is added below:

Is 0 a Real Number?

In the world of mathematics, understanding zero's classification is crucial as it influences various aspects of arithmetic and algebra. 0 is a real number as Real numbers include all rational and irrational numbers. Rational numbers are those that can be expressed as a fraction of two integers. 0 can be expressed as [Tex]\frac{0}{1}[/Tex]​, which fits this definition.

Also, Since 0 can be written as 01\frac{0}{1}10​, it is a rational number, and hence, a real number.

In the number system, real numbers include all the numbers that can be found on the number line, encompassing integers, fractions, and irrational numbers. Since 0 can be expressed as a fraction (0/1) and lies on the number line, it fits the criteria for real numbers.

After considering all of the facts stated above we can confirm that 0 is a real number.

Also Read:

• Is 0 a Natural Number?
• Is 0 a rational or irrational number?
• Is zero a rational number?

Types of Real Numbers

Real numbers are divided into two main types:

1. Rational Numbers (ℚ): Numbers that can be expressed as a fraction (p/q) where both the numerator represented as p and the denominator represented as q are integers, and the denominator (q) is not zero. Rational numbers include integers, finite decimals, and repeating decimals (e.g., 1/2, -3, 0.75).
2. Irrational Numbers: Numbers that cannot be expressed in the form of a simple fraction p/q where 'p' and 'q' are integers and the denominator 'q' is not equal to zero (q≠0) and have non-terminating, non-repeating decimal expansions. They cannot be represented as a fraction of two integers (e.g., √2, πぱい).

Set of Real Numbers

All the numbers known to humans except the imaginary numbers come under the set of the real number. We use R to denote the set of real numbers. We can have various subsets of the real number that denote different types of numbers. Various subsets of the Real number are,

Real Numbers can be divided into the following subsets:

List of Real Numbers

The list of real numbers is endless because it includes all kinds of numbers like whole, natural, integers, rational, and irrational numbers. Since it includes integers it has negative numbers too. So, there is no specific number from which the list of real numbers starts or ends. It goes to infinity towards both sides of the number line.

Symbol of Real Numbers

We use R to represent a set of Real Numbers and other types of numbers can be represented using the symbol discussed below,

Properties of Real Numbers

There are different properties of Real numbers with respect to the operation of addition and multiplication, which are as follows:

Learn More: Properties of Numbers

Real Numbers on a Number Line

A number line contains all the types of numbers like natural numbers, rational numbers, Integers, etc. As shown in the following number line 0 is present in the middle of the number line. Positive integers are written on the right side of zero whereas negative integers are written on the left side of zero, and there are all possible values in between these integers.

Rational numbers are written between the numbers they lie. For example, 3/2 equals to 1.5, so is noted between 1 and 2. It shows that the number 3/2 lies somewhere between 1 and 2.

Similarly, the Number 13/4 = 3.25 lies between 3 and 4. So we noted it between 3 and 4. Number -50/9 = -5.555. . . , lies between -5 and -6. So we noted it between -5 and -6 on the number line.

Example: Represent the Following numbers on a number line:

• 23/5
• 6
• -33/7

Solution:

The rational numbers,

• 23/5
• 6
• -33/7

can easily be represented in a number line as,

Irrational Numbers on Number Line

Irrational Numbers can't be represented on the number line as it is, we need clever tricks and geometry to represent irrational numbers on a number line.

Learn More : Representation of √3 on a number line

Decimal Expansion of Rational Numbers

The decimal expansion of a real number is its representation in base equals to 10 (i.e., in the decimal system). In this system, each "decimal place" consists of a digit from 0 to 9. These digits are arranged such that each digit is multiplied by a power of 10, decreasing from left to right.

Let’s Expand 13/4

So 13/4 can also be written as 3.25.

Now Let's take another example. Let’s expand 1/3

So 1/3 can also be written as 0.3333…… We can also write it as [Tex]0.\overline3 [/Tex]

Similarly, 1/7 can be written as 0.142857142857142857... or [Tex]0.\overline{142857}  [/Tex]. This is known as the recurring decimals expansion.

Decimal Expansion of Irrational Numbers

Decimal Expansion of Irrational Numbers is non-terminating and non-repeating. We can find the decimal expansion such as √2, √3, √5, etc. using the long division method. The decimal Expansion of √2 is up to three digits after the decimal is calculated in the following illustration.

Learn More: Square root of 2

Solved Examples Problems on Real Numbers

Here are some example problems on Real Numbers and their properties.

Example 1: Add √3 and √5

Solution:

(√3 + √5)

Now answer is an irrational number.

Example 2: Multiply √3 and √3.

Solution:

√3 × √3 = 3

Now answer is a rational number.

So we can say that result of mathematical operations on irrational numbers can be rational or irrational.

Now add a rational number with an irrational number.

Example 3: Add 2 and √5

Solution:

(2 + √5)

Now answer is an irrational number.

People Also Read:

• Numbers
• Natural Numbers
• Whole Numbers
• Integers
• Rational Numbers
• Irrational numbers

Real Numbers Class 10 - Resources

Real Numbers is a very important topic for class 10. Here are the resources where you can check NCERT solutions and notes for real numbers.

• Real Numbers Class 10 Notes
• Real Numbers Class 10 NCERT Solutions

Also Read: How to Teach Real Numbers

Practice Problems on Real Numbers

1. Add √2 and √8.

2. Multiply √7 by √14.

3. Add 5 to √9.

4. Add 3/2 (a rational number) to √3 (an irrational number)

5. Classify the Following Numbers as Rational or Irrational:

• a) [Tex]\sqrt{49}[/Tex]
• b) [Tex]\pi[/Tex]
• c) [Tex]0.75[/Tex]
• d) [Tex]\sqrt{2}[/Tex]

6 Find the LCM and HCF of 24 and 36.

7. Express 0.3333… (repeating) as a Fraction.

8. Prove that [Tex]\sqrt{3}[/Tex] is an Irrational Number.

9. Find the Decimal Representation of [Tex]\frac{7}{8}[/Tex]. Is it Terminating or Non-Terminating?

10. Find the HCF of 75 and 105 using the Euclidean Algorithm.

11. Write the Prime Factorization of 420 and 252. Use it to Find their LCM.

12. Is 0.1010010001… a Rational or Irrational Number? Justify your Answer.

Conclusion

Real numbers provide the basic foundation for many mathematical concepts, ranging from all natural numbers to complex irrational numbers. They provide a complete framework to deal with quantities, measurements, and relations of objects in discipline mathematics. Be it basic arithmetic or highly advanced algebraic operations, properties and features of the real numbers are important in carrying out one's analysis and solving problems. features of the real numbers are essential in carrying out one's analysis and solving problems. Understanding the whole structure of real numbers, representation, and decimal expansion enhances mathematical knowledge and makes it applicable to many real-life situations.