Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Find the modulus \(r\) of the number. Do not use a calculator. $$-24+7 i$$

Step-by-Step Solution

Answer

The modulus of the complex number is 25.

Step 1: Understand the Definition

The modulus of a complex number \(a + bi\) is given by \( r = \sqrt{a^2 + b^2} \). Here, \(a = -24\) and \(b = 7\). The modulus measures the 'distance' of the complex number from the origin in the complex plane.

Step 2: Square the Real Part

We first square the real part of the complex number: \[ (-24)^2 = 576 \]

Step 3: Square the Imaginary Part

Next, we square the imaginary part of the complex number: \[ (7)^2 = 49 \]

Step 4: Sum the Squares

Add the squares of the real and imaginary parts:\[ 576 + 49 = 625 \]

Step 5: Take the Square Root

The last step is to take the square root of the sum obtained in step 4 to find the modulus: \[ r = \sqrt{625} = 25 \]

Key Concepts

Modulus of a Complex Number

The modulus is a very important concept when dealing with complex numbers. It provides a way to measure the 'size' or 'length' of a complex number. Think of it as the distance from the origin to a point in the complex plane.
The complex number is written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The formula to find the modulus \(r\) is \(r = \sqrt{a^2 + b^2}\).
In our example, the complex number is \(-24 + 7i\). By plugging \(a = -24\) and \(b = 7\) into the formula, we get the modulus \(r = \sqrt{(-24)^2 + (7)^2}\). This tells us how far the complex number is from the origin. Hence, it's very similar to finding the hypotenuse in a right-angled triangle on the complex plane.

• The modulus gives the absolute magnitude of the complex number.
• It's always a non-negative number.

Real Part of a Complex Number

The real part of a complex number is the component that does not involve the imaginary unit \(i\). In a complex number \(a + bi\), the "\(a\)" is the real part. If we have a complex number like \(-24 + 7i\), then the real part is \(-24\).
This value can be thought of as the horizontal axis position of the point representing the complex number on the complex plane.
It's important to note that:

• The real part can be positive, negative, or zero.
• It affects the modulus calculation through the term \(a^2\) in the modulus formula \(r = \sqrt{a^2 + b^2}\).

Identifying the real part is crucial for visualizing where the complex number sits along the horizontal axis as well as for calculations involving complex conjugates.

Imaginary Part of a Complex Number

The imaginary part of a complex number is represented by the term involving \(i\), where \(i\) is the imaginary unit defined as \(i^2 = -1\).
For a complex number \(a + bi\), the imaginary part is \(b\). Using our given example, \(-24 + 7i\), the imaginary part here is \(7\).
This component tells you how far up or down the vertical axis the complex number is in the complex plane.

• The imaginary part can also be any real number (positive, negative, or zero).
• It significantly impacts the modulus calculation with the term \(b^2\) in \(r = \sqrt{a^2 + b^2}\).

Understanding the imaginary part helps visualize the position on the vertical part of the complex plane, influencing the overall position of the number.

Complex Plane

The complex plane is a two-dimensional plane used to graphically represent complex numbers. It is very similar to the coordinate plane used in algebra. In this plane, the horizontal axis (x-axis) is used for the real part, while the vertical axis (y-axis) stands for the imaginary part.
A complex number \(a + bi\) corresponds to the point \((a, b)\) on the complex plane. For instance, the complex number \(-24 + 7i\) is represented as the point \((-24, 7)\) on this plane.

• Points in the complex plane show both real and imaginary components.
• The origin represents the complex number \(0 + 0i\).
• The distance from the origin to the point \((a, b)\) is the modulus, \(r = \sqrt{a^2 + b^2}\).

This graphical approach makes it easy to visually understand operations with complex numbers, such as addition, multiplication, and finding moduli.