Immerse yourself in the fascinating realm of** mathematics** as we journey through the intriguing concept of **comparing** and **contrasting** the **absolute values** of a **real number** compared to that of a **complex number** system.

This article will illuminate these principles, comparing and contrasting the **absolute value function** as it applies to **real** and **complex numbers**.

**Definition**

The **absolute value** of a number is a measure of its magnitude, regardless of its sign. This concept applies to both **real** and **complex numbers**, though it manifests a bit differently in each case due to their inherent properties.

**Absolute Value of a Real Number**

For a **real number**, the **absolute value**, denoted as |x|, represents its distance from zero on the number line. If x is positive, |x| = x. If x is negative, |x| = -x. So, the absolute value is always a** non-negative real number**.

For example, **|-3| = 3** and** |4| = 4**.

**Absolute Value (or Modulus) of a Complex Number**

The concept of **absolute value** extends to **complex numbers** but is often referred to as the **modulus** in this context. For a complex number** z = a + bi** (where** a** and **b** are real numbers, i is the imaginary unit), the **modulus**, denoted as **|z|**, is defined as the distance from the origin to the point representing the complex number in the complex plane.

It’s calculated using the **Pythagorean theorem**:

|z| = √(a² + b²)

So, the **modulus** is also a **non-negative real number**.

For example, if **z = 3 + 4i**, then **|z| = √(3² + 4²) = 5**.

**Properties**

**Non-negativity**

#### Real Numbers

The **absolute value** of a **real number** is always **non-negative**. This is because it is defined as the **distance** from zero on the number line, and distance can’t be **negative**.

#### Complex Numbers

The **modulus** of a **complex number** is also always **non-negative**. This is because it’s defined as the **distance** from the origin in the **complex plane**, which also can’t be negative.

**Identity of Indiscernibles**

#### Real Numbers

**|x| = 0** if and only if **x = 0**.

#### Complex Numbers

**|z| = 0** if and only if **z = 0**.

**Multiplicative**

#### Real Numbers

The **absolute value** has a **multiplicative property**, i.e., **|xy| = |x||y|** for all real numbers **x** and **y**.

#### Complex Numbers

The **modulus** has the same **multiplicative property**, i.e., **|zw| = |z||w|** for all complex numbers **z** and **w**.

**Triangle Inequality**

#### Real Numbers

The **absolute value** of a **real number** obeys the **triangle inequality**: **|x + y| ≤ |x| + |y|** for all real numbers **x** and **y**.

#### Complex Numbers

The **modulus** of a **complex number** also obeys the **triangle inequality**: **|z + w| ≤ |z| + |w|** for all complex numbers **z** and **w**.

**Conjugate Property**

#### Real Numbers

The **absolute value** of a **real number** is equal to the absolute value of its negative, i.e., **|x| = |-x|**.

#### Complex Numbers

The **modulus** of a **complex number** is equal to the modulus of its conjugate, i.e., **|z| = |z̅|** where** z̅** denotes the conjugate of **z**.

**Squared Modulus and Dot Product**:

#### Real Numbers

For **real numbers**, **squaring** the **absolute value** of a number gives the square of the number itself: **|x|² = x²**.

#### Complex Numbers

For **complex numbers**, **squaring** the **modulus** of a **complex number** gives the dot product of the complex number with its conjugate: **|z|² = z * z̅.** This doesn’t have an equivalent in real numbers because real numbers don’t have conjugates.

**Dimensions**

#### Real Numbers

The **absolute value** of a real number deals with a **one-dimensional number line**.

#### Complex Numbers

The **modulus** of a **complex number** works in** two dimensions**, the **complex plane**, incorporating both the** real** and **imaginary** parts of the **complex number**.

**Exercise**

**Example 1**

**Real Numbers**

Let’s take a negative **real number** **-7**.

Absolute Value:

| -7 | = 7

This is because the absolute value of a real number is the non-negative value of the number, irrespective of its sign.

**Example 2**

**Complex Numbers**

Now, let’s move to **complex numbers**. Take a complex number **z = 3 + 4i**.

Modulus:

| z | = √(3² + 4²)

| z | = 5

The modulus of a complex number is the square root of the sum of the squares of its real and imaginary parts.

### To Compare and Contrast

For the

**real numbers**, the**absolute value**is just the number itself if it is**positive**and the number without its sign if it is**negative**. The concept of**absolute value**is straightforward in**one dimension**, as it simply gives the**distance**from zero.For

**complex numbers**, however, the**modulus**gives the distance from the origin in the complex plane. This value incorporates both the**real**and**imaginary**parts of the complex number, which makes it a**more complex**concept (no pun intended) compared to the absolute value for real numbers.

The absolute value and the modulus, although calculated differently, both provide a measure of** “size”** or **“magnitude”** for real and **complex numbers,** respectively.

**Applications **

**Physics**

In the field of **quantum mechanics**, complex numbers are often used to represent **quantum states**, and the **modulus** of the complex number is used to find the probability of a quantum system being found in a particular state.

**Engineering**

In **electrical engineering**, **complex numbers** are used to analyze **AC circuits**. The **modulus** of a complex number can represent the **amplitude** of a wave (such as voltage or current), while the **angle** represents the **phase shift**.

**Computer Science**

In **digital signal processing** and **image processing**, **complex numbers** are often used. The **modulus** of the complex number can be used to determine the **magnitude** of the signal or pixel intensity, while the **angle** can represent **phase** or **orientation**.

**Mathematics**

In pure** mathematics**, **complex numbers** and their **modulus** are used in a wide variety of areas, including **geometry**, **calculus**, and **number theory**. The comparison of **absolute values** and **modulus** can help to build a deeper understanding of these mathematical structures.

**Economics and Data Analysis**

**Absolute values** of **real numbers** are often used in **economics** and** data analysis** to measure deviations or differences. For instance, when comparing **forecasted** and **actual data**, the **absolute value** of the difference can give a measure of the forecast error.

**Geography and Geology**

The **absolute value** is used to determine **distances** or differences in **elevations**, which are always considered as **positive quantities**.