Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Find the modulus \(r\) of the number. Do not use a calculator. $$6+8 i$$

Step-by-Step Solution

Answer

The modulus is 10.

Step 1: Identify the Complex Number

The given complex number is \( 6 + 8i \), where \( 6 \) is the real part and \( 8 \) is the imaginary part coefficient.

Step 2: Understand Modulus Formula

The modulus \( r \) of a complex number \( a + bi \) is given by the formula \( r = \sqrt{a^2 + b^2} \). Here, \( a = 6 \) and \( b = 8 \).

Step 3: Substitute into the Formula

Substitute \( a = 6 \) and \( b = 8 \) into the modulus formula: \( r = \sqrt{6^2 + 8^2} \).

Step 4: Calculate the Squares

Calculate \( 6^2 = 36 \) and \( 8^2 = 64 \).

Step 5: Sum the Squares

Add the results from Step 4: \( 36 + 64 = 100 \).

Step 6: Find the Square Root

Take the square root of the sum from Step 5: \( r = \sqrt{100} = 10 \).

Key Concepts

Modulus of a Complex Number

To understand the modulus of a complex number, think of it as the distance from the origin to the point representing the complex number on the complex plane. A complex number takes the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The modulus, often symbolized by \(|z|\), is calculated using the formula \(r = \sqrt{a^2 + b^2}\). This formula comes from the Pythagorean theorem, as complex numbers can be visualized as right triangles.

In the given problem, the complex number is \(6 + 8i\). To find its modulus, identify \(a = 6\) and \(b = 8\), and apply these values to the modulus formula:

• Calculate the squares: \(6^2 = 36\) and \(8^2 = 64\).
• Add them to get \(36 + 64 = 100\).
• Finally, take the square root to find \(r = \sqrt{100} = 10\).

This means the modulus of \(6 + 8i\) is 10.

Imaginary Numbers

Imaginary numbers are at the heart of complex number theory. They arise when solving equations that involve the square roots of negative numbers, which do not have solutions in the set of real numbers. The imaginary unit, denoted by \(i\), is defined such that \(i^2 = -1\).

In any complex number like \(a + bi\), the \(b\) coefficient represents the imaginary part. Here, \(bi\) signifies that \(b\) times \(i\) provides the imaginary component. This allows complex numbers to handle operations that involve the roots of negatives and gives them the ability to represent a wider scope of values than real numbers alone.

In the exercise, \(8i\) indicates that 8 is the imaginary part, thus making it easy to differentiate it from the real part \(6\). Understanding imaginary numbers is essential in fields that involve equations unsolvable by real numbers alone, such as electrical engineering and quantum physics.

Real Part and Imaginary Part

Breaking down a complex number, it consists of two key components: the real part and the imaginary part. These are often written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Understanding these parts helps when performing arithmetic operations on complex numbers or when expressing them graphically. The real part, \(a\), lies on the horizontal axis, while the imaginary part, \(bi\), lies on the vertical axis of the complex plane.

For \(6 + 8i\), \(6\) is the real part and \(8\) is the imaginary part. In solving problems, clearly distinguishing these two parts is crucial. For instance, when calculating the modulus, each part is squared separately, their sum is obtained, and then the square root of their sum gives the modulus. This detailed yet clear separation of real and imaginary components is a key aspect of working with complex numbers.