Real Numbers

Real numbers are numbers that show us the distance on a line. Any number that can be placed on this line is called as a real number. These numbers numbers can be positive or negative, like: 

Whole numbers (0, 1, 2, 3,...)

Integers (-3, -2, -1, 0, 1, 2,..)

Fractions (½, -¾) 

Decimals (0.75, -2.3)

Irrational numbers (π ≈ 3.14159 or 2 ≈ 1.414)

The numbers that are not rational are called imaginary or complex numbers, and they are not real numbers. Like, for example, -1, 2 + 3i, and -i. 

A number line is a visual representation of real numbers. It helps us see how numbers are arranged, including whole numbers, fractions, decimals, and even irrational numbers. Let’s see the visual representation of a number line.

In order to represent real numbers on a number line, follow these simple steps:

Step 1: You must draw a horizontal line with arrows on both the ends (which denotes that the number line is never ending). 

Step 2: We know that the number 0 (zero) is the origin of the numbering system, so put number 0 in the middle of the number line. 

Step 3: Mark the positive numbers on the right side of the number line and negative numbers on the left side of the number line.

Step 4: The line is completed by irrational numbers filling in the spaces between rational numbers, which occupy the points at a finite distance.

Real numbers include all the numbers we use in everyday life, such as whole numbers, fractions, decimals, and irrational numbers. The real number chart helps us understand how those different types of numbers are related to each other.

Real Numbers (R)
│
├── Rational Numbers (Q) (Can be written as a fraction p/q)
│   │
│   ├── Integers (Z) (..., -3, -2, -1, 0, 1, 2, 3, ...)
│   │   │
│   │   ├── Whole Numbers (W) (0, 1, 2, 3, 4, ...)
│   │   │   │
│   │   │   ├── Natural Numbers (N) (1, 2, 3, 4, ...)
│   │   │
│   │   ├── Negative Integers (-1, -2, -3, -4, ...)
│   │
│   ├── Fractions (1/2, -3/4, 5/8, ...)
│   │
│   ├── Terminating Decimals (0.5, 0.25, 0.75, ...)
│   │
│   ├── Repeating Decimals (0.333..., 0.666..., 1.272727..., ...)
│
├── Irrational Numbers (Q) (Cannot be written as a fraction p/q)
│   │
│   ├── π (3.1415926535...)
│   ├── √2 (1.414213562...)
│   ├── e (2.718281828...)
│   ├── Golden Ratio (1.618033988...)

Real numbers include different types of numbers that we use in our everyday life and mathematics. They can be classified into rational numbers that can be written as fractions, and irrational numbers that cannot be written as simple fractions. Let’s understand them in detail.

Rational Numbers: A rational number is any number that can be written as a fraction in the form of pq, where p and q are integers, and q  0. This means rational numbers include whole numbers, integers, and fractions. Examples of rational numbers are ½, -3, 4, and 0.75 (which is ¾). 

Irrational Numbers: An irrational number is any number that cannot be written as a simple fraction pq, where p and q are integers, and p  0. These numbers have non-repeating, non-terminating decimal expansions. Examples of irrational numbers include π  3.14159…, 2 1.414…, and e  2.718. 

The letter R represents real numbers, which include both rational numbers (Q) and irrational numbers (Q). Thus, we can write a set of real numbers

 R = Q ⋃ Q.

In the numbering system, the number categories are represented using different symbols or alphabets. Let’s look at them:

N - Natural numbers

W - Whole numbers

Z - Integers

Q - Rational numbers

Q - Irrational numbers

Real numbers follow fundamental mathematical properties or rules. They are closure, associativity, commutativity, and distributive properties. Below are some key properties of real numbers that define their behavior in mathematical operations. 

Closure Property: It states that when you add or multiply two real numbers, the result is always a real number. That is, if a and b are two real numbers, such that 

    
            a + b = R

            a × b = R

Associative Property: It states that when adding or multiplying three real numbers, the result stays the same no matter how the numbers are grouped. That is, if a, b, c are real numbers, then 

a + (b + c) = (a + b) + c 

a × (b × c) = (a × b) + c  

Commutative Property: It states that, the sum and the product of two real numbers stay the same even when the order of the number is changed. That is, if a,  are real numbers, then

            a + b = b + a

Distributive Property: It states that multiplication distributes over addition and subtraction. That is, 

Multiplication over addition  a × (b + c) = (a × b) + (a × c)

Multiplication over subtraction  a × (b – c) = (a × b) – (a × c)

Real numbers are used in many everyday situations, from counting objects to measuring distances and temperatures. Here are some real life examples of real numbers:

• Temperature: One of the best examples to show integers are the temperature reading in degrees. A cold winter day, for example, might show – 5° C on a thermometer. A thermometer has number readings from negative to positive values. 
   
• Bank Transactions: In a bank, a deposit means adding money and is shown with a positive value, like +50 dollars credited. On the other hand, withdrawal of money is represented by negative signs. Like, for example, – 50 dollars debited. 
   
• Sharing or dividing things: If you eat ¾ of a whole pizza, ¼ is left. This is an example of using a rational number. 
   
• Measurements: A ruler or measuring tape shows real numbers from 0 to 100 with fractions or decimal numbers in between each integer.