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Absolute Value Of A Complex Number

Q: Is the absolute value of a complex number always a real number?

Yes, always. The formula \\sqrt{a^2 + b^2}) takes two real components (\a) and )), squares and adds them, and takes the square root. The result of this process is always a non-negative real number because it represents a physical distance.

Q: How do you find the absolute value of a pure imaginary number like \5i)?

You use the same formula. For \z = 5i), you can write it as \0 + 5i). So, \a=0) and =5). The absolute value is \\sqrt{0^2 + 5^2} = \sqrt{25} = 5). The absolute value of any pure imaginary number i) is simply \|b|).

Q: What does the absolute value of the difference of two complex numbers, \|z_1 - z_2|), represent?

This is a more advanced but very useful concept. Just as \|x-y|) is the distance between two numbers on the number line, \|z_1 - z_2|) represents the distance between the points for \z_1) and \z_2) on the complex plane. It's a way to find the distance between any two complex numbers, not just from the origin.

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Ever wondered how to measure the 'size' of a number that has both a real and an imaginary part? This lesson will guide you through the absolute value of complex numbers, connecting it to concepts you already know like distance and the Pythagorean theorem.

Absolute Value Of A Complex Number — an original Algebra911 reference diagram defining absolute value of a complex number with its key formula and a worked example. — Absolute Value of a Complex Number

What Is the Absolute Value of a Complex Number?

The absolute value of a complex number is its distance from the origin $(0, 0)$ on the complex plane. For a complex number written in the standard form $z = a + b i$ , its absolute value is denoted as $| z |$ and is always a non-negative real number. This concept extends the idea of absolute value from the one-dimensional number line to the two-dimensional complex plane. Another name for the absolute value of a complex number is the modulus. The terms 'absolute value' and 'modulus' are used interchangeably in this context.

Think of it like this: if you were to draw a point on a graph representing your complex number, the absolute value is the length of the straight line connecting that point directly to the center of the graph (the origin). It's a measure of magnitude or 'size', telling you how far the number is from zero, just in two dimensions instead of one.

How Does This Relate to the Absolute Value of Real Numbers?

You're likely already familiar with the absolute value of real numbers. For any real number $x$ , its absolute value $| x |$ is its distance from zero on the number line. For example, $| 5 |$ is $5$ because it's $5$ units away from zero. Similarly, $| - 5 |$ is also $5$ because it's also $5$ units away from zero. Distance is always a positive concept.

This definition is perfectly consistent with complex numbers. A real number like $5$ can be written as a complex number with a zero imaginary part: $5 + 0 i$ . A number like $- 5$ can be written as $- 5 + 0 i$ . If we apply the complex number rule (which we'll define shortly), we'll see that the concept holds perfectly. The absolute value of a complex number is a natural extension of this 'distance from zero' idea, moving it from a simple line into a two-dimensional plane.

Why Is the Complex Plane So Important for This?

The complex plane is the key to visualizing and truly understanding the absolute value of a complex number. Unlike real numbers that live on a one-dimensional number line, complex numbers have two parts: a real part ( $a$ ) and an imaginary part ( $b$ ). To graph them, we need two axes.

The horizontal axis is the real axis, just like a regular number line.
The vertical axis is the imaginary axis, used to plot the imaginary part.

A complex number $z = a + b i$ is plotted as the point $(a, b)$ . For instance, the number $z = 3 + 4 i$ corresponds to the point $(3, 4)$ on the complex plane. You go $3$ units to the right on the real axis and $4$ units up on the imaginary axis.

Once you plot this point, you can draw a line segment from the origin $(0, 0)$ to your point $(a, b)$ . The length of this line segment is precisely the absolute value, $| z |$ . Looking at the graph, you might notice that this line segment forms the hypotenuse of a right-angled triangle. The other two sides of the triangle are the real part ( $a$ ) and the imaginary part ( $b$ ). This visual connection is what leads us directly to the famous theorem we use for the calculation.

How Do You Calculate the Absolute Value of a Complex Number?

Calculating the absolute value of a complex number $z = a + b i$ involves a direct application of the Pythagorean theorem. As we saw on the complex plane, the distance from the origin to the point $(a, b)$ forms a right triangle with legs of length $| a |$ and $| b |$ .

The Pythagorean theorem states that for a right triangle with legs $l e g_{1}$ and $l e g_{2}$ and hypotenuse $c$ , we have $l e g_{1}^{2} + l e g_{2}^{2} = c^{2}$ . In our case, the legs are the real component $a$ and the imaginary component $b$ , and the hypotenuse is the absolute value $| z |$ . So, we have:

a^{2} + b^{2} = | z |^{2}

To find $| z |$ , we simply take the square root of both sides. Since distance cannot be negative, we only consider the positive root. This gives us the fundamental formula for the absolute value of a complex number.

|z| = \sqrt{a^2 + b^2}

Here is a step-by-step process to follow every time:

Identify the real and imaginary parts. For a complex number $z = a + b i$ , identify the values of $a$ and $b$ . Be careful with signs!
Square both parts. Calculate $a^{2}$ and $b^{2}$ . Remember that squaring a negative number results in a positive number.
Add the results. Sum the two squared values: $a^{2} + b^{2}$ .
Take the square root. Calculate the square root of the sum to find the absolute value: $\sqrt{a^{2} + b^{2}}$ .

Notice that the imaginary unit $i$ itself is not part of the calculation. We only use the real coefficient $b$ from the imaginary part.

Let's Work Through Some Examples

Example 1

Find the absolute value of the complex number $z = 3 + 4 i$ .

Step 1: Identify $a$ and $b$ .
In $z = 3 + 4 i$ , the real part is $a = 3$ and the imaginary part is $b = 4$ .

Step 2: Square both parts.
$a^{2} = 3^{2} = 9$
$b^{2} = 4^{2} = 16$

Step 3: Add the results.
$a^{2} + b^{2} = 9 + 16 = 25$

Step 4: Take the square root.
$| z | = \sqrt{25} = 5$

So, the absolute value of $3 + 4 i$ is $5$ . This means the point $(3, 4)$ is $5$ units away from the origin on the complex plane.

Example 2

Calculate the modulus of $z = - 5 - 12 i$ .

Step 1: Identify $a$ and $b$ .
For $z = - 5 - 12 i$ , we have $a = - 5$ and $b = - 12$ . It's crucial to include the negative signs.

Step 2: Square both parts.
Remember that squaring a negative number gives a positive result.
$a^{2} = (- 5)^{2} = 25$
$b^{2} = (- 12)^{2} = 144$

Step 3: Add the results.
$a^{2} + b^{2} = 25 + 144 = 169$

Step 4: Take the square root.
$| z | = \sqrt{169} = 13$

The modulus of $- 5 - 12 i$ is $13$ .

Example 3

Find the absolute value of the pure imaginary number $z = - 7 i$ .

Step 1: Identify $a$ and $b$ .
This number can be written as $z = 0 - 7 i$ . So, the real part is $a = 0$ and the imaginary part is $b = - 7$ .

Step 2: Square both parts.
$a^{2} = 0^{2} = 0$
$b^{2} = (- 7)^{2} = 49$

Step 3: Add the results.
$a^{2} + b^{2} = 0 + 49 = 49$

Step 4: Take the square root.
$| z | = \sqrt{49} = 7$

The absolute value of $- 7 i$ is $7$ . This makes sense, as the point $(0, - 7)$ is exactly $7$ units away from the origin along the imaginary axis.

Key formulas for absolute value of a complex number by Algebra911.

What Are Some Common Mistakes to Avoid?

When first learning to calculate the absolute value of complex numbers, students often make a few common errors. Being aware of these can help you avoid them. Let's look at the calculation for $z = 8 - 6 i$ , where the correct answer is $| z | = \sqrt{8^{2} + (- 6)^{2}} = \sqrt{64 + 36} = \sqrt{100} = 10$ .

Common Mistake	Incorrect Calculation	Why It's Wrong
Forgetting to Square a Negative $b$ Correctly	$\sqrt{8^{2} - 6^{2}} = \sqrt{64 - 36} = \sqrt{28}$	The formula is $a^{2} + b^{2}$ . The value of $b$ is $- 6$ , and $(- 6)^{2}$ is $+ 36$ , not $- 36$ . The sum inside the square root should never be smaller than the largest squared component.
Including the Imaginary Unit $i$	$\sqrt{8^{2} + (- 6 i)^{2}} = \sqrt{64 + 36 i^{2}} = \sqrt{64 - 36} = \sqrt{28}$	The formula uses $b$ , the real coefficient of the imaginary part, not $b i$ . The absolute value must be a real number, so $i$ should never be part of the formula itself.
Adding Before Squaring	$\sqrt{(8 + (- 6))^{2}} = \sqrt{2^{2}} = \sqrt{4} = 2$	The formula requires you to square the components first and then add them. This is a common order of operations mistake. $a^{2} + b^{2}$ is not the same as $(a + b)^{2}$ .
Forgetting the Square Root	$8^{2} + (- 6)^{2} = 64 + 36 = 100$	Forgetting the final step is easy to do. $100$ is the value of $\| z \|^{2}$ , not $\| z \|$ . The absolute value is a distance, and $100$ is far too large for the point $(8, - 6)$ .

Quick Reference and Summary

Here are the most important takeaways from this lesson condensed into a quick summary. Use this as a reference when you're doing practice problems.

Definition: The absolute value of a complex number, also called the modulus, is its distance from the origin on the complex plane.
Notation: The absolute value of $z$ is written as $| z |$ .
Key Concept: It's the length of the hypotenuse of a right triangle whose legs are the real part ( $a$ ) and the imaginary part ( $b$ ).
The Formula: For any complex number $z = a + b i$ , the formula is:

|z| = \sqrt{a^2 + b^2}

The Result: The absolute value of a complex number is always a non-negative, real number.
The Process:

Identify $a$ and $b$ .
Square them: $a^{2}$ and $b^{2}$ .
Add them: $a^{2} + b^{2}$ .
Take the square root: $\sqrt{a^{2} + b^{2}}$ .

Frequently Asked Questions

What is the difference between 'modulus' and 'absolute value'?

For complex numbers, there is no difference. The terms 'modulus' and 'absolute value' mean the exact same thing: the distance from the number to the origin on the complex plane. You can use them interchangeably.

Is the absolute value of a complex number always a real number?

Yes, always. The formula $\sqrt{a^{2} + b^{2}}$ takes two real components ( $a$ and $b$ ), squares and adds them, and takes the square root. The result of this process is always a non-negative real number because it represents a physical distance.

Can the absolute value of a complex number be negative?

No, it cannot. Since the absolute value represents a distance, it must be either positive or zero. The only complex number with an absolute value of $0$ is $0 + 0 i$ , which is the origin itself.

How do you find the absolute value of a pure imaginary number like

5 i

You use the same formula. For $z = 5 i$ , you can write it as $0 + 5 i$ . So, $a = 0$ and $b = 5$ . The absolute value is $\sqrt{0^{2} + 5^{2}} = \sqrt{25} = 5$ . The absolute value of any pure imaginary number $b i$ is simply $| b |$ .

Why don't you include the 'i' in the formula?

The formula is based on the Pythagorean theorem using the coordinates of the point on the complex plane, which are $(a, b)$ . These coordinates are real numbers that represent distances along the axes. The 'i' is just a marker to tell us that $b$ is the vertical component, but the distance itself is the real number $b$ .

Does the absolute value tell you the 'value' of the number?

It tells you the 'magnitude' or 'size' of the number, but not its direction. Two different complex numbers, like $3 + 4 i$ and $5$ , can both have the same absolute value of $5$ . This means they are both the same distance from the origin, but in different locations on the complex plane.

What does the absolute value of the difference of two complex numbers,

| z_{1} - z_{2} |

, represent?

This is a more advanced but very useful concept. Just as $| x - y |$ is the distance between two numbers on the number line, $| z_{1} - z_{2} |$ represents the distance between the points for $z_{1}$ and $z_{2}$ on the complex plane. It's a way to find the distance between any two complex numbers, not just from the origin.