Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Find the modulus \(r\) of the number. Do not use a calculator. $$3-4 i$$

Step-by-Step Solution

Answer

The modulus of 3 - 4i is 5.

Step 1: Identify Components

To find the modulus of a complex number, identify the real and imaginary parts. The given complex number is \(3 - 4i\), where the real part \(a\) is 3 and the imaginary part \(b\) is -4.

Step 2: Apply the Modulus Formula

The modulus \(r\) of a complex number \(a + bi\) is given by the formula \(r = \sqrt{a^2 + b^2}\). For the complex number \(3 - 4i\), apply this formula by substituting \(a = 3\) and \(b = -4\).

Step 3: Compute Squares of Components

Calculate \(a^2\) and \(b^2\). Here, \(3^2 = 9\) and \((-4)^2 = 16\).

Step 4: Add the Squares

Add the squares of the components: \(9 + 16 = 25\).

Step 5: Take the Square Root

Find \(r\) by taking the square root of the sum: \(r = \sqrt{25} = 5\).

Key Concepts

Modulus of a Complex Number

When dealing with complex numbers, we often need to find their modulus or absolute value. The modulus gives us an idea of the size or magnitude of the complex number in the complex plane. For a complex number of the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, the formula to calculate the modulus \( r \) is:

• \( r = \sqrt{a^2 + b^2} \)

This formula essentially applies the Pythagorean Theorem, where \( a \) and \( b \) form the legs of a right triangle, and \( r \) is the hypotenuse, or the distance from the origin to the point \( (a, b) \) on the complex plane.

By applying this formula to a specific example, such as \( 3 - 4i \), we can calculate the modulus without needing a calculator by following these steps:

• Identify \( a = 3 \) and \( b = -4 \).
• Calculate \( a^2 = 3^2 = 9 \) and \( b^2 = (-4)^2 = 16 \).
• Add the squares: \( 9 + 16 = 25 \).
• Take the square root: \( r = \sqrt{25} = 5 \).

Therefore, the modulus of \( 3 - 4i \) is 5.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a fundamental concept in complex numbers. The imaginary unit is defined by the property:

• \( i^2 = -1 \)

This definition enables us to extend real numbers into the complex number system, providing solutions to equations that are otherwise unsolvable with real numbers alone. For example, the equation \( x^2 + 1 = 0 \) has no real solutions because the square of a real number cannot be negative. However, using \( i \), the solutions become \( x = i \) and \( x = -i \).

In a complex number, the imaginary part is the coefficient of \( i \). For the complex number \( 3 - 4i \), the imaginary part is \(-4\), and \( i \) facilitates its combination with real numbers. This imaginary unit expands the number system into complex numbers, represented as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, making it possible to perform a wide range of mathematical operations.

Complex Plane

The complex plane is an essential concept for visualizing and understanding complex numbers. It is a two-dimensional plane where each point represents a complex number. The horizontal axis is known as the real axis, and the vertical axis is the imaginary axis.

Every complex number \( a + bi \) corresponds to a point \( (a, b) \) on this plane. Here, \( a \) is the distance from the origin along the real axis, and \( b \) is the distance along the imaginary axis. This intuitive graphical representation helps us comprehend operations with complex numbers, such as addition, multiplication, and the calculation of modulus.

For example, the number \( 3 - 4i \) can be plotted on the complex plane at the point \( (3, -4) \). This point is 3 units across the real axis and 4 units down the imaginary axis. Understanding the complex plane not only enhances the comprehension of complex numbers but also allows for deeper exploration into more advanced topics like complex conjugates and mapping functions. By equating complex numbers with points or vectors in this plane, we gain a powerful tool to visualize mathematical concepts.