Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The absolute value is 8.

Step 1: Understanding the Exercise

The problem asks us to find the absolute value of a complex number. In this case, we have the complex number \(8i\), which is purely imaginary.

Step 2: Recall the Formula for Absolute Value of a Complex Number

The absolute value of a complex number \(a + bi\) is given by the formula \(\sqrt{a^2 + b^2}\). In our problem, \(a = 0\) and \(b = 8\).

Step 3: Apply the Formula

Substitute the values of \(a\) and \(b\) into the formula: \[\text{Absolute value} = \sqrt{0^2 + 8^2} = \sqrt{64}\]

Step 4: Simplify the Expression

Now simplify \(\sqrt{64}\) to find the absolute value:\[\sqrt{64} = 8\]

Step 5: Write the Final Answer

The absolute value of the complex number \(8i\) is \(8\).

Key Concepts

Absolute Value

The concept of absolute value is quite significant when dealing with both real and complex numbers. Here, absolute value refers to the non-negative value of a number without considering its direction. When it comes to complex numbers, the absolute value is sometimes called the modulus. It represents the distance of the complex number from the origin on the complex plane. For a real number, the absolute value is simply the number itself if it's positive, and the opposite if it's negative.

• For example, the absolute value of \(3\) is \(3\), and the absolute value of \(-3\) is also \(3\).

When you see an absolute value symbol, like \(|8i|\), think of it as distance. For complex numbers in particular, this concept helps compare magnitudes without getting caught up with direction. It's always a positive value or zero, never negative.

Imaginary Numbers

Imaginary numbers were developed to solve equations that didn't have solutions in the real number system, such as the square root of negative numbers. The fundamental imaginary unit is denoted by \(i\), and its defining property is that \(i^2 = -1\).
This concept extends the idea of numbers from the one-dimensional number line to the two-dimensional complex plane, where numbers have both a "real" part and an "imaginary" part. Any complex number can be expressed in the form \(a + bi\) where \(a\) is the real part and \(b\) is the imaginary part.

Here are some key points regarding imaginary numbers:

• Imaginary numbers alone (like \(8i\)) sit vertically on the complex plane.
• They are broadening our numerical perspective rather than just existing for the sake of odd math. Think of them as new dimensions, just like adding width adds to length to create a more complete picture.

Formula for Absolute Value of Complex Numbers

The formula for finding the absolute value of a complex number is wonderfully straightforward and is mostly about applying the Pythagorean theorem.
To determine the absolute value of a complex number written as \(a + bi\), use the formula: \[|a + bi| = \sqrt{a^2 + b^2}\]This formula calculates the distance from the origin on the complex plane to the point representing the complex number. In essence, you're using the familiar concept of Euclidean distance to measure how far away a point is from the origin.

• If the complex number is purely real, such as \(5\), its absolute value is \(\sqrt{5^2} = 5\).
• If the number is purely imaginary, like in our example \(8i\), you calculate it via \(\sqrt{0^2 + 8^2} = 8\).

This formula underlies much of complex analysis and problem solving in math and engineering, helping you handle not just position, but magnitude without the clutter of direction.