Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Find the modulus \(r\) of the number. Do not use a calculator. $$2-3 i$$

Step-by-Step Solution

Answer

The modulus is \(\sqrt{13}\).

Step 1: Understand the Problem

We need to find the modulus of the complex number \(2 - 3i\). The modulus \(r\) of a complex number \(a + bi\) is calculated as \(r = \sqrt{a^2 + b^2}\).

Step 2: Identify the Real and Imaginary Parts

For the complex number \(2 - 3i\), the real part \(a\) is 2 and the imaginary part \(b\) is -3.

Step 3: Square the Real and Imaginary Parts

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

Step 4: Sum the Squares

Add the squares from the previous step: \(a^2 + b^2 = 4 + 9 = 13\).

Step 5: Compute the Square Root

Find the square root of the sum calculated: \(r = \sqrt{13}\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The 'i' represents the imaginary unit, which is the square root of -1. Complex numbers are essential because they help in solving equations that don't have real solutions, enabling deeper understanding in fields like engineering and physics.

• The real part is the number without the 'i', and it behaves like a regular number on the number line.
• The imaginary part is the part that includes 'i', which stands for \(\sqrt{-1}\).

Together, these two parts work collaboratively to provide a richer number system that extends beyond the limitations of the real number line. When finding the modulus, or magnitude, of a complex number, we are essentially discovering its length from the origin on the complex plane.

Imaginary Unit

The imaginary unit, denoted by ‘i’, is an essential concept in understanding complex numbers. It is defined as \(i = \sqrt{-1}\). This definition means that \(i^2 = -1\). By introducing 'i', mathematicians allowed the square root of negative numbers to be expressed in a meaningful way.

• In the expression \(2 - 3i\), the '-3i' signifies that 3 parts of the number are along the imaginary axis.
• The multiplication by 'i' transforms the real number into an imaginary one.

Mastering the concept of 'i' helps in performing operations on complex numbers, such as multiplication, division, and finding the modulus. Without 'i', handling the square root of negative numbers would have remained an unsolvable issue, thereby leaving many mathematical and engineering problems unresolved.

Square Root Calculation

The square root calculation is crucial when determining the modulus of a complex number. To find the modulus of a complex number \(a + bi\), you use the formula \(r = \sqrt{a^2 + b^2}\). This formula is derived from the Pythagorean theorem and represents the distance between the point \((a, b)\) on the complex plane and the origin \((0, 0)\).

• First, square the real component \(a\) and the imaginary component \(b\) separately.
• Next, add these squares together.
• Finally, take the square root of this sum, which gives the modulus of the complex number.

In our example with the complex number \(2 - 3i\), we calculated \(a^2 = 4\) and \(b^2 = 9\). Adding them gives us 13, and the square root yields the modulus \(\sqrt{13}\). Understanding this calculation is fundamental to working with the magnitude of complex numbers and their representation on the complex plane.