Real Numbers

In mathematics, real numbers represent a huge set of numbers that includes both rational and irrational numbers. From integers to fractions and irrational values, real numbers cover different subsets that are highly important to perform important algebraic operations in both mathematics and ordinary life. Understanding real numbers builds a strong foundation for algebra, geometry, calculus, and information interpretation. This article explains real number in detail along with their properties, types, representation, real-life examples and solved problems.

Table Of Contents

• What are Real Numbers?
• Subset of Real Numbers
• Symbol for Real Number
• Properties of Real Numbers
• Importance of Properties of Real Numbers
• Real Number Chart
• List of Real Numbers
• Real-Life Examples of Real Numbers
• Conclusion
• FAQs on Real Numbers

 

What are Real Numbers?

Real numbers denoted by letter R, comprise of integers, rational numbers (fractions and decimals), and irrational numbers (like 2 or π). All these numbers can be represented on the number line. It doesn't consists of imaginary numbers or unreal numbers that can't be represented using number line. Given below are some examples and representation of real numbers.

For example:

• 5 is a huge variety.

• −3.6 is the real quantity.

• √2 is likewise a real range.

• π (pi) is, without a doubt, a big range.

 

Subset of Real Numbers

The Real Numbers form a big set, and the smaller devices' internal are called subsets of the real numbers:

• Natural Numbers: Counting numbers beginning from 1, 2, 3, …

• Whole Numbers: Natural numbers plus zero (0, 1, 2, …)

• Integers: Negative and wonderful whole numbers which incorporate zero (… −3, −2, −1, zero, 1, 2,3 …)

• Rational Numbers: Numbers that can be expressed within the form p/q, wherein q ≠ 0 (e.g.,1/2, 3, 5).

• Irrational Numbers: Numbers that are not written as a clean fraction (e.g., π, 2, 5)

Each of those styles of real numbers plays a role in helping us describe portions and measurements in several contexts. 

Symbol for Real Number

The image for the real huge range is ℝ. In arithmetic, we write:

• ℝ: All Real Numbers

• ℚ: Rational Numbers

• ℝ  ℚ: Irrational Numbers

• ℤ: Integers

• ℕ: Natural Numbers
  

 

Properties of Real Numbers

The four main properties of real numbers are:

• Commutative Property
• Associative Property 
• Distributive Property
• Identity Property

Commutative Property

Definition:

The commutative property states that the order of the numbers does not have an impact on the end result of addition or multiplication of real numbers. For example, if a and b are two real numbers then,

Addition: a + b = b + a

Multiplication: a × b = b × a

Example:

• Addition: 2 + 5 = 5 + 2 = 7

• Multiplication: 3 × 6 = 6 × 3 = 18

Explanation:

This proves that even if you change the order of numbers while performing addition and multiplication on any real numbers the result is the same real number. This gives flexibility in calculations, allowing numbers to be rearranged without changing the final results.

 

Associative Property

Definition: The associative property states that the way numbers are grouped in addition or multiplication does not impacts the end result.

For addition: (a + b) + c = a + (b + c)

For multiplication: (a × b) × c = a × (b × c)

Example:

(2 + 3) + 4 = 2 + (3 + 4) = 9

(1 × 2) × 3 = 1 × (2 × 3) = 6

Explanation:

This permits simplification of expressions by regrouping the numbers. It gives flexibility in calculations, by permitting rearrangement of numbers without impacting the final result.

 

Distributive Property

Definition:

The distributive property applies to multiplication and addition operation. It states that multiplying a number with sum is addend is similar to multiplying the range through every addend and then adding the outcomes. That is, for any real numbers a, b and c:

a × (b + c) = ab + ac

Example:

2 × (3 + 4) = 2 × 3 + 2 × 4 = 6 + 8 = 14

Explanation:

This property is important in algebra for representation of expressions and equations.

Identity Property

Definition:

There are identity properties of addition and multiplication. Identity elements are numbers that if used in addition or multiplication keep the number same.

Additive Identification: 0 (a + 0 = a)

Multiplication  Identification: 1 (a × 1 = a)

Example:

• 8 + 0 = 8
• 9 × 1 = 9

Explanation:

These elements function as key elements in operations and are essential in solving equations.

 

Inverse Elements

Definition:

Inverse elements are numbers that reverse the impact of addition or multiplication. There are two types of inverse elements: opposite of x i.e., (−x) and reciprocal of x i.e., (1/x)

Additive inverse: for each a, the additive inverse is a because a (−a) = 0. 

Multiplicative inverse: For every non-zero a, the multiplicative inverse is 1/a, whilst you endure in thoughts that a × (1/a) = 1.

Example:

6 + (−6) = zero

4 × (1/4) = 1

Explanation:

These inverses are used to undo operations; it is vital in solving equations and algebraic functions.

 

Table of properties

 

Importance of Properties of Real Numbers

Real numbers properties are highly important in mathematics. They help us in everyday work to get predictable outcomes of calculations. Whether working on equations, simplifying expressions, or factoring them, these properties form the foundation of most mathematical operations.

 

Real Number Chart

This real number chart summarises the properties of different types of real numbers.

 

List of Real Numbers

Here’s a list of real numbers for better clarity:

• Positive integers: 1, 2, 3, 4, …

• Negative integers: −1, −2, −3, …

• Zero: 0

• Rational numbers: 1/2, −3/4, 0.75, 2.0

• Irrational numbers: π, √2, e
  

 

Real-Life Examples of Real Numbers

Let’s learn about the real-life examples of real numbers:

• Bank Balances

If you deposit ₹1000 and withdraw ₹1500, your account suggests −₹500. The balance shown in accoount is an example of a Real Number (mainly an integer).

• Temperature

A thermometer displaying −5°C in cold climate and 38.5°C in summers. Both minus and decimal numbers here are examples of Real Numbers.

• Measuring Heights

Your height may be 5.6 feet or 102.2 cm. These decimal values are examples of Real Numbers used in normal lifestyles.

• Geometry and Construction

The value of π (about 3.1416) is applied in calculating the location of circles. π is an irrational number and thus is an example of real number.

Conclusion

By now, you have a strong understanding of what real numbers are, along with their types and properties. You have explored the special types of real numbers, such as rational and irrational numbers, and discovered how they work within the real number system. You’ve also familiarised with ways to read and write real numbers and can now identify and classify the subsets of real numbers. Through real-life examples like temperature readings and bank transactions, you've seen how real numbers are used in our daily life scenarios. By practising more about examples  based on real numbers you will build a strong foundation required for algebraic operations involving real numbers.

Frequently Asked Questions on Real Numbers

1. What is the concept of real numbers?

Answer: Real numbers include all rational and irrational numbers that can be represented on the number line.

2.  What are the properties of real numbers?

Answer: The four main properties of real numbers are:

1. Commutative Property
2. Associative Property 
3. Distributive Property
4. Identity Property

3. Why are real numbers called real?

Answer: They are known as "real" to distinguish them from "imaginary" numbers such as −1.

4. Is pi a real number?

Answer: Yes, π is an irrational natural number.

Explore the world of real numbers and master their concepts with ease by starting your learning journey with Orchids International School now!