Problem 11 - Contemporary Precalculus

Question

In Exercises \(9-14,\) find the absolute value. $$|1+\sqrt{2} i|$$

Step-by-Step Solution

Answer

Answer: The absolute value of the complex number \(1+\sqrt{2}i\) is \(\sqrt{3}\).

Step 1: Identify the real and imaginary parts of the complex number

The given complex number is \(1 + \sqrt{2}i\). Here, the real part (a) is 1, and the imaginary part (b) is \(\sqrt{2}\).

Step 2: Apply the formula for the absolute value of a complex number

To find the absolute value of a complex number \(a + bi\), use the formula: \(|a + bi| = \sqrt{a^2 + b^2}\). In our case, a = 1 and b = \(\sqrt{2}\).

Step 3: Plug in the values and calculate the absolute value

Plug in the values of a and b into the formula: \(|1 + \sqrt{2}i| = \sqrt{1^2 + (\sqrt{2})^2} = \sqrt{1 + 2} = \sqrt{3}\) So, the absolute value of the given complex number is \(\sqrt{3}\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have two parts: 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 imaginary unit \(i\) has a unique property where \(i^2 = -1\), which helps in handling operations involving square roots of negative numbers.

• Complex numbers expand the one-dimensional number line into a two-dimensional plane.
• They are essential in various fields, including engineering, physics, and mathematics.
• Complex numbers allow us to work with concepts such as oscillations and waves.

Complex numbers can help solve problems that have no solutions in the realm of real numbers alone.

Real and Imaginary Parts

Every complex number has a real part and an imaginary part. When you look at a complex number in the form \(a + bi\), \(a\) represents the real part, and \(b\) is the coefficient of the imaginary part \(i\).

• Real part \(a\) determines the displacement along the horizontal axis in the complex plane.
• Imaginary part \(b\) affects the movement along the vertical axis.
• Together, they form a coordinate on the complex plane, also known as the Argand plane.

Understanding how to separate these two components is crucial when performing arithmetic and finding the absolute value of a complex number.

Formula for Absolute Value

The absolute value of a complex number, also called its magnitude, measures the "distance" from the origin to that point in the complex plane. It is analogous to the distance formula we use in geometry.To find the absolute value of a complex number \(a + bi\), you use the formula: \[|a + bi| = \sqrt{a^2 + b^2}\]

• The formula involves squaring both the real part \(a\) and the imaginary part \(b\).
• Adding these squares gives a positive value since both terms are squared.
• Taking the square root transforms this sum into a non-negative value representing the magnitude of the complex number.

This formula is pivotal for solving exercises involving complex numbers.

Step by Step Solution

To solve the exercise and find the absolute value of the given complex number \(1 + \sqrt{2}i\), we follow these steps:**Step 1: Identify the real and imaginary parts.**

• The real part \(a\) is 1.
• The imaginary part \(b\) is \(\sqrt{2}\).

**Step 2: Use the formula for absolute value.**Substitute the values of \(a\) and \(b\) into the formula:\[|a + bi| = \sqrt{1^2 + (\sqrt{2})^2}\]**Step 3: Calculate the absolute value.**Perform the arithmetic within the formula:

• Calculate \(1^2 = 1\).
• Calculate \((\sqrt{2})^2 = 2\).
• Add these results: \(1 + 2 = 3\).
• Take the square root: \(\sqrt{3}\).

Hence, the absolute value of the complex number \(1 + \sqrt{2}i\) is \(\sqrt{3}\).