In **mathematics**, few concepts possess the **real number system**‘s captivating power and profound significance. From the enchanting elegance of **whole numbers** to the mystifying infinitude of **decimals**, the **real number system** is the bedrock upon which mathematical analysis is built.

While we may encounter **real numbers** daily, their true nature and remarkable properties often remain concealed beneath the surface. In this article, we embark on a journey of discovery to demystify the **real number system**, unravel its intricacies, and appreciate its boundless wonders.

So, fasten your intellectual seatbelts and prepare to explore the captivating universe of **real numbers**.

**Defining Real Number System**

The **real number system** is a fundamental mathematical construct encompassing the set of all possible numbers, **rational** and **irrational**, and serves as the foundation for mathematical analysis.

**Real numbers** are characterized by their ability to represent quantities on a **continuous number line**, where each point corresponds to a unique **real number**.

The **real number system** includes familiar entities such as **whole numbers**, **integers**, **fractions**, **decimals**, less intuitive numbers like **square roots**, **transcendental numbers** (e.g., **π** and **e**), and even non-repeating decimals.

This expansive collection of numbers exhibits crucial properties, such as **closure** under **addition**, **subtraction**, **multiplication**, and** division**, enabling intricate mathematical investigations and providing a framework for modeling and understanding the **physical world**.

Figure-1.

**Properties**** of Real Number System**

The **real number system** possesses several key properties that define its nature and make it a fundamental **mathematical construct**. Let’s delve into these properties in detail:

### Closure Property

The **real number system** is **closed** under **addition** and **multiplication**. This means that when you add or multiply two **real numbers**, there is always another **real number**. For example, if **a** and **b** are **real numbers**, then **a + b** and **a * b** are **real numbers**.

### Commutative Property

The **operations** of **addition** and **multiplication** are **commutative** in the **real number system**. This means that changing the order of the numbers added or multiplied does not affect the result.

For instance, for **real numbers** **a** and **b**, **a + b** = **b + a** and **a * b** = **b * a**. The **commutative property** allows for **flexibility** in rearranging terms without altering the outcome of the operation.

### Associative Property

**Addition** and **multiplication** in the **real number system** are **associative operations**. This means that the **grouping** of added or multiplied numbers does not affect the result.

For any **real numbers** **a**, **b**, and **c**, **(a + b) + c** = **a + (b + c)** and **(a * b) * c** = **a * (b * c)**. The **associative property** ensures that the order in which operations are performed does not change the outcome.

### Distributive Property

The **real number system** obeys the **distributive property**, which relates **addition** and **multiplication**. For **real numbers** **a**, **b**, and **c**, the distributive property states that **a * (b + c)** = **a * b + a * c**.

### Identity Elements

The **real number system** possesses two **identity elements** – one for **addition** and **multiplication**. The **additive identity** is **0**, which means that for any **real number** **a**, **a + 0 = 0 + a = a**. The **multiplicative identity** is **1**, so for any **real number** **a**, **a * 1 = 1 * a = a**.

### Inverse Elements

Every **non-zero real number** has an **additive inverse** and a **multiplicative inverse**. The **additive inverse** of a real number **a** is denoted as **-a** and satisfies the property **a + (-a) = (-a) + a = 0**. The **multiplicative inverse** of a non-zero real number **a** is denoted as **1/a** or **a^(-1)** and satisfies the property **a * (1/a) = (1/a) * a = 1**.

### Ordering Property

The **real number system** is equipped with an **ordering relation**, which allows us to compare and arrange **real numbers** in ascending or descending order. This property is known as the **trichotomy law**. For any two **real numbers**, **a** and **b**, exactly one of the following holds: **a < b** (a is less than b), **a > b** (a is greater than b), or **a = b** (a is equal to b).

### Density Property

The **real number system** exhibits the **density property**, which states that between any two distinct **real numbers**, there exists another **real number**. This implies that there are **infinitely many real numbers** between any two given real numbers, making the **real number line** densely populated.

### Completeness Property

The **real number system** possesses the **completeness property**, the **Dedekind completeness**, or the **least upper bound property**. This property guarantees that every non-empty set of **real numbers** bounded above has a **least upper bound** (supremum) in the real number system. It ensures no “gaps” or “holes” in the **real number line**, allowing for precise **mathematical analysis** and **continuity**.

**Applications **

The **real number system** finds applications in various fields and disciplines as a fundamental **mathematical construct**. Here are some examples of how the real number system is applied in different fields:

### Physics and Engineering

The **real number system** is **essential** for **modeling and analyzing physical phenomena** in **physics** and **engineering**. **Real numbers** represent **distance**, **time**, **temperature**, **energy**, and more. The principles of **calculus**, which heavily rely on **real numbers**, are employed to **describe and predict** the behavior of **physical systems**.

### Economics and Finance

**Real numbers** are extensively used in **economics** and **finance** to represent quantities such as **prices**, quantities of **goods and services**, **interest rates**, **investments**, and **monetary values**. Concepts like **supply and demand**, **market equilibrium**, and **financial calculations** rely on real numbers for accurate analysis and decision-making.

### Computer Science and Information Technology

In **computer science** and **information technology**, real numbers are employed for **numerical computations**, **algorithms**, **simulations**, and **data analysis**. Real numbers represent and process **continuous data**, such as **sensor measurements**, **image and signal processing**, and **scientific simulations**.

### Statistics and Data Analysis

Real numbers play a crucial role in **statistics** and **data analysis**, where they are used to represent **numerical data**, measure **variability**, calculate **averages**, and perform **statistical tests**. Real numbers enable the interpretation and communication of **quantitative information** derived from data.

### Architecture and Design

The real number system is utilized in **architecture** and **design** to express and work with **measurements**, **dimensions**, **proportions**, and **spatial relationships**. Architects and designers rely on real numbers to ensure **accuracy** and **precision** in creating buildings, structures, and artistic compositions.

### Medicine and Biotechnology

Real numbers are applied in **medicine** and **biotechnology** for **medical measurements**, **data analysis**, **drug dosage calculations**, and **modeling physiological processes**. Real numbers enable accurate representation and interpretation of **biological** and **medical data**, facilitating diagnoses, treatment plans, and research.

### Geographic Information Systems (GIS)

**GIS systems** extensively utilize real numbers for representing **spatial coordinates**, **distances**, **elevations**, and other **geographic attributes**. Real numbers enable precise mapping, navigation, location-based services, and spatial analysis.

### Cryptography and Cybersecurity

**Real numbers** are involved in **cryptographic algorithms**, which provide secure communication and data protection. Real numbers play a role in **mathematical operations** used in encryption, decryption, **digital signatures**, and **secure key generation**.

### Probability Theory and Risk Analysis

**Real numbers** are employed in **probability theory** and **risk analysis** to quantify uncertainties and assess probabilities. Real numbers enable the formulation of **probability distributions**, **risk models**, and **statistical measures** used in decision-making and risk management.

**Exercise **

### Example 1

Simplify the expression:** √(9 + √16)**

### Solution

We start by simplifying the expression inside the square root:

= √(9 + √16)

= √(9 + 4)

= √13

The simplified expression is **√13**.

### Example 2

Solve the equation: **|2x – 3| = 7**

Figure-2.

### Solution

We consider two cases: when 2x – 3 is positive and negative.

#### Case 1: 2x – 3 > 0

In this case, 2x – 3 = 7. Adding 3 to both sides, we get 2x = 10. Dividing by 2, we find x = 5.

#### Case 2: 2x – 3 < 0

In this case, we have -(2x – 3) = 7. Expanding the negative sign, we get -2x + 3 = 7. Subtracting 3 from both sides, we have -2x = 4. Dividing by -2, we find x = -2.

The solutions to the equation |2x – 3| = 7 are: **x = 5** and **x = -2**.

### Example 3

Find the **domain** of the function **f(x) = √(5 – x)**.

Figure-3.

### Solution

The domain of a function is the set of all possible values of** x** for which the function is defined. In this case, the square root function is defined for non-negative real numbers.

Therefore, the expression inside the square root must be non-negative:

5 – x ≥ 0

Solving the inequality, we find:

x ≤ 5.

The **domain** of the function **f(x) = √(5 – x)** is **x ≤ 5**.

### Example 4

Evaluate the expression: **log₁₀(1000)**

### Solution

The logarithm functions** log₁₀(x)** gives the power to which** 10** must be raised to obtain **x**. In this case, we have **log₁₀(1000)= 3** since **10³ = 1000**.

The value of **log₁₀(1000)** is **3**.

### Example 5

Solve the** inequality**: **2x – 5 > 3x + 1**

### Solution

To solve the inequality, we need to isolate **x** on one side.

2x – 5 > 3x + 1

Subtracting **2x** from both sides, we get:

-5 > x + 1

Subtracting **1** from both sides, we have:

-6 > x

The solution to the inequality **2x – 5 > 3x + 1** is **x < -6**.

### Example 6

Find the absolute value of **-8**.

### Solution

The absolute value of a **real number** is its distance from zero on the number line. The absolute value of **-8** is **8**, as** -8** is **8** units away from zero in the **negative direction**.

The absolute value of **-8** is **8**.

### Example 7

Find the **decimal representation** of the fraction **3/5**.

### Solution

To find the decimal representation of the fraction **3/5**, we divide **3** by** 5**.

3 ÷ 5 = 0.6

The decimal representation of the fraction **3/5** is **0.6**.

### Example 8

Solve the equation: **2x² – 5x + 2 = 0**.

### Solution

We can solve this **quadratic equation** by factoring, completing the **square**, or using the **quadratic formula**.

Using factoring: The equation factors as **(2x – 1)(x – 2) = 0**. Setting each factor equal to zero, we have two possibilities:

2x – 1 = 0

which gives:

x = 1/2

x – 2 = 0

which gives

x = 2

The solutions to the equation **2x² – 5x + 2 = 0** are **x = 1/2** and **x = 2**.