Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-10: Find the absolute value. $$ |5+8 i| $$

Step-by-Step Solution

Answer

The absolute value is \( \sqrt{89} \).

Step 1: Understand the Modulus Formula for Complex Numbers

The absolute value of a complex number, also called the modulus, is calculated using the formula: \[ |a + bi| = \sqrt{a^2 + b^2} \]. In this problem, you need to find the absolute value of \( 5 + 8i \), so identify \( a = 5 \) and \( b = 8 \).

Step 2: Plug Values into the Modulus Formula

Substitute \( a = 5 \) and \( b = 8 \) into the modulus formula: \[ |5 + 8i| = \sqrt{5^2 + 8^2} \].

Step 3: Calculate the Squares

Calculate the squares of \( a \) and \( b \):\( 5^2 = 25 \) and \( 8^2 = 64 \).

Step 4: Sum the Square Results

Add the results of the squared terms: \( 25 + 64 = 89 \).

Step 5: Find the Square Root of the Sum

Calculate the square root of the sum: \( \sqrt{89} \).

Step 6: Final Result

The absolute value of the complex number \( 5 + 8i \) is \( \sqrt{89} \).

Key Concepts

Understanding Absolute Value in Complex Numbers

The absolute value in the context of complex numbers represents the distance from the origin to the point, placed in the complex plane, representing the complex number. This can be visualized as the length of the line segment from the origin \(0 + 0i\) to the complex number \(a + bi\). \(
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• The absolute value is essentially a measure of the magnitude of the complex number.
• It is always a non-negative number.

Calculating the absolute value involves using a specific formula derived from the Pythagorean theorem. We square the real part and the imaginary part separately, sum them up, and finally, take the square root. This step ensures that the result conveys the concept of distance, which is why no negative results arise.

Modulus of a Complex Number

The modulus of a complex number is another term for what we often call the absolute value. This term emphasizes the geometric interpretation of "length" or "size" in the complex number space. \(
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• The formula used to find the modulus is written as \[ |a + bi| = \sqrt{a^2 + b^2} \].
• It identifies the distance from the point \(a + bi\) to the origin in the complex plane.

For example, if you are tasked to find the modulus of \(5 + 8i\), you will first recognize \(a = 5\) and \(b = 8\). Substitute these into the formula to find the output, which represents the modulus of the number.

Operations on Complex Numbers

Complex number operations are fundamental when handling complex equations or expressions. They allow us to perform mathematical actions that include addition, subtraction, multiplication, and division, just like we do with real numbers, albeit with additional rules when considering the imaginary unit \(i\). \(
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• Addition/Subtraction: Combine real parts with real parts and imaginary parts with imaginary parts.
• Multiplication: Distribute and use the identity \(i^2 = -1\) to simplify.
• Division: Multiply by the complex conjugate to eliminate the imaginary part in the denominator.

These operations ensure that we can manipulate complex expressions just like any numeric values. Understanding them is essential when dealing with problems that involve multiple complex numbers.

Square Roots in Complex Numbers

The concept of square roots is different when it comes to complex numbers. While in typical arithmetic, square roots are tied to non-negative numbers, complex numbers transcend this limitation. Any negative number can have a square root in the realm of complex numbers due to \(i\), where \(i = \sqrt{-1}\). \(
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• Taking the square root of a negative number involves moving into the complex plane using \(i\).
• For example, \(\sqrt{-4}\) equals \(2i\) since \(\sqrt{-4} = \sqrt{4} \times \sqrt{-1}\).

Using the square root in the modulus formula of complex numbers permits us to interpret results as distances or lengths, a practical way to quantify their magnitude. Complex numbers extend our ability to operate fluently with square roots that involve negative values, offering broader avenues for problem-solving.