Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Find the modulus \(r\) of the number. Do not use a calculator. $$12-5 i$$

Step-by-Step Solution

Answer

The modulus is 13.

Step 1: Identify Real and Imaginary Parts

Start by identifying the real and imaginary parts of the complex number. For the complex number \(12 - 5i\), the real part is \(12\) and the imaginary part is \(-5\).

Step 2: Use Modulus Formula

The modulus \(r\) of a complex number \(a + bi\) is given by the formula \(r = \sqrt{a^2 + b^2}\). In our case, \(a = 12\) and \(b = -5\).

Step 3: Calculate Squares of Real and Imaginary Parts

Calculate \(a^2\) and \(b^2\). For this example: \(12^2 = 144\) and \((-5)^2 = 25\).

Step 4: Sum the Squares

Add \(a^2\) and \(b^2\) to get the sum: \(144 + 25 = 169\).

Step 5: Find the Square Root

Find the square root of the sum. \(\sqrt{169} = 13\).

Key Concepts

Modulus of a Complex Number

The modulus of a complex number is like finding the "distance" from the origin in the complex plane. Imagine a point on a graph, where the horizontal axis (x-axis) represents the real numbers and the vertical axis (y-axis) represents the imaginary numbers.
When you have a complex number like \(12 - 5i\), it corresponds to a point \((12, -5)\) on the complex plane.

• Think of the modulus as the hypotenuse of a right triangle, where the real and imaginary parts are the two legs.
• The formula for the modulus is: \[ r = \sqrt{a^2 + b^2}\]where \(a\) is the real part and \(b\) is the imaginary part.

In this case, by substituting \(a = 12\) and \(b = -5\), you calculate the modulus as \(r = 13\). This is the length of the line from the origin to the point \((12, -5)\). It gives you an idea of how "far" the complex number is from zero.

Real and Imaginary Parts

In a complex number, the real part and the imaginary part are crucial components that tell us how complex numbers are formed. Any complex number \(a + bi\) consists of:

• Real Part: This is the number \(a\). It's the component of the complex number that lies on the real number line, parallel to the x-axis on the complex plane. In our example, the real part is \(12\).
• Imaginary Part: This is the number \(b\) that multiplies with \(i\), the imaginary unit. \(i\) is defined by the property \(i^2 = -1\). The imaginary part places the complex number on the y-axis. Here, it's \(-5\).

The combination of these two parts allows complex numbers to be visualized as points in a two-dimensional plane, richening the scope beyond the usual one-dimensional real number line. Understanding real and imaginary parts is fundamental before calculating any related properties like modulus.

Square Root Calculation

Square root calculation is essential in finding the modulus of a complex number. Let's go over why it's important, especially when tackling problems without a calculator.To get the modulus, you first take the sum of the squares of the real and imaginary parts. For the complex number \(12 - 5i\):

• The real part squared is \(12^2 = 144\).
• The imaginary part squared is \((-5)^2 = 25\).
• Add these squares: \(144 + 25 = 169\).

Once you have this sum, finding the square root gives you the modulus: \(\sqrt{169} = 13\). Understanding how to find square roots, especially common perfect squares, can make these calculations simple.
The ability to compute square roots without a calculator is a valuable skill, helping in exams or any scenario where quick mental math is required.