# Compare and Contrast the Absolute Value of a Real Number to That of a Complex Number – Definition and Examples

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 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.