Problem 12 - Contemporary Precalculus

Question

In Exercises \(9-14,\) find the absolute value. $$|2-3 i|$$

Step-by-Step Solution

Answer

Answer: The absolute value of the complex number \(2 - 3i\) is \(\sqrt{13}\).

Step 1: Identify the real and imaginary parts

We identify the real part, \(a\), and the imaginary part, \(b\), of the given complex number, \(2 - 3i\). In this case, the real part is \(a = 2\) and the imaginary part is \(b = -3\).

Step 2: Use the absolute value formula

Now using the formula \(|a + bi| = \sqrt{a^2 + b^2}\) with \(a = 2\) and \(b = -3\), we have: $$ |2 - 3i| = \sqrt{2^2 + (-3)^2} $$

Step 3: Solve

Calculate the sum of the squares in the radical: $$ |2 - 3i| = \sqrt{4 + 9} = \sqrt{13} $$ Therefore, the absolute value of \(2-3i\) is: $$ |2 - 3i| = \sqrt{13} $$

Key Concepts

Absolute Value of Complex Numbers

The absolute value of a complex number gives us an idea of its magnitude on the complex plane. For a complex number of the form \(a + bi\), the absolute value is calculated using the formula \(|a + bi| = \sqrt{a^2 + b^2}\). This formula derives from the Pythagorean theorem, which finds the hypotenuse of a right triangle.
In the case of the complex number \(2 - 3i\), we calculate as follows:

• Identify the real part \(a = 2\) and the imaginary part \(b = -3\).
• Apply the formula: \(|2 - 3i| = \sqrt{2^2 + (-3)^2}\).
• Calculate \(2^2 = 4\) and \((-3)^2 = 9\), and hence, \(\sqrt{4 + 9} = \sqrt{13}\).

Therefore, the absolute value \(|2 - 3i|\) equals \(\sqrt{13}\). Understanding this concept helps us measure the distance from the origin to the point \( (2, -3) \) on the complex plane.

Imaginary Numbers

Imaginary numbers are an essential component of complex numbers. They are numbers that, when squared, yield a negative result. The foundation of imaginary numbers lies in the imaginary unit, denoted by \(i\), which is defined as \(i = \sqrt{-1}\). When squared, \(i^2 = -1\).
Imaginary numbers allow us to extend the real number system and solve equations that don't have solutions in the real numbers alone, such as \(x^2 + 1 = 0\).
In our example, the expression \(-3i\) is the imaginary part, where \(-3\) is multiplied by \(i\). Imaginary numbers, along with real numbers, help define complex numbers in the form \(a + bi\). Understanding imaginary numbers is crucial for working with complex numbers and their various applications in mathematics and engineering.

Complex Plane

The complex plane, sometimes known as the Argand plane, is a graphical representation of complex numbers. It resembles the Cartesian coordinate system but interprets numbers differently.
On this plane:

• The horizontal axis (known as the real axis) represents the real part of complex numbers.
• The vertical axis (known as the imaginary axis) represents the imaginary part.

By plotting a complex number \(a + bi\), we place it at the point \((a, b)\) on this plane. For instance, the complex number \(2 - 3i\) positions the point at \((2, -3)\).
Visualizing complex numbers on the complex plane allows students and mathematicians to understand operations like addition, subtraction, and the geometric interpretation of the magnitude of complex numbers, which is precisely the "absolute value" discussed earlier. The complex plane thus serves as an indispensable tool for analyzing and manipulating complex numbers.

Mathematical Formula

A mathematical formula is a concise way of expressing information symbolically, such as a relation between different quantities. When dealing with complex numbers, formulas help simplify and solve problems efficiently.
For complex numbers, the absolute value formula \(|a + bi| = \sqrt{a^2 + b^2}\) is particularly useful. It translates the geometric idea of distance in the complex plane into an algebraic form.
Using mathematical formulas:

• Helps clearly set out relationships and properties of numbers and operations
• Provides a step-by-step method for solving complex mathematical problems
• Makes calculations more efficient and accurate

In the context of our problem, applying the formula \(|2 - 3i| = \sqrt{2^2 + (-3)^2} = \sqrt{13}\) efficiently provides the magnitude of the complex number \(2 - 3i\). Understanding how to use and apply these formulas is an essential skill in mathematics, aiding in a more profound understanding that can be applied in various fields.