Problem 5 - Precalculus with Limits

Question

In Exercises 5-10, plot the complex number and find its absolute value. \(-6 + 8i\)

Step-by-Step Solution

Answer

The complex number \(-6+8i\) is plotted at the coordinates (-6,8) on the complex plane. Its absolute value is 10.

Step 1: Plotting the complex number

Complex numbers are plotted on the complex plane in a similar way to (x,y) coordinates in the Cartesian system. The real part represents the x-coordinate and the imaginary part represents the y-coordinate. So, for \(-6+8i\), plot a point at (-6,8).

Step 2: Finding the absolute value of the complex number

The absolute value of a complex number, \(\sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts of the complex number respectively, represents its distance from the origin. In this case, \(a = -6\) and \(b = 8\). Substitute these values into the equation and compute the absolute value. This will yield \(\sqrt{(-6)^2 + 8^2} = \sqrt{100} = 10\)

Key Concepts

Complex Plane

The complex plane is a unique mathematical concept used to represent complex numbers graphically. It is similar to the Cartesian coordinate system but designed to handle complex numbers. In this plane:

• The horizontal axis, known as the real axis, represents the real component of complex numbers.
• The vertical axis, known as the imaginary axis, represents the imaginary component.

Each complex number is located at a specific point in this plane via coordinates (real, imaginary). For instance, the complex number \(-6 + 8i\) can be found at the point (-6, 8). This visual representation significantly helps in understanding operations with complex numbers, such as addition, subtraction, and finding distances.

Absolute Value

The absolute value (or modulus) of a complex number provides a measure of its distance from the origin in the complex plane. For a complex number such as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, its absolute value is calculated using the formula: \[|a + bi| = \sqrt{a^2 + b^2}.\]This formula stems from the Pythagorean theorem, reflecting how complex numbers are inherently linked to two-dimensional geometry. For example, the absolute value of \(-6 + 8i\) is determined by plugging in the values into the formula: \(\sqrt{(-6)^2 + 8^2} = \sqrt{100} = 10\). This tells us the straight-line distance from the origin (0,0) to the point (-6,8) on the complex plane.

Imaginary Part

The imaginary part of a complex number is a crucial component that sets it apart from real numbers. Given a complex number \(a + bi\), the imaginary part is \(b\) and is associated with the imaginary unit \(i\), defined as the square root of \(-1\). Imaginary numbers, and by extension the imaginary part of complex numbers, arise naturally in contexts where equations involve the square root of negative numbers.
In the case of \(-6 + 8i\), the imaginary part is \(8\). When plotted on the complex plane, this component gives the vertical position of the number, differentiating it from the real component, which affects the horizontal position.

Real Part

The real part of a complex number resembles conventional numbers we frequently encounter. For any complex number \(a + bi\), the real part is simply \(a\). It defines the complex number's position along the real axis on the complex plane.
This component allows complex numbers to interact with real-world mathematical operations, such as addition or multiplication with other real numbers. In the example \(-6 + 8i\), the real part is \(-6\). This places the complex number negatively along the real axis, influencing its x-coordinate in the complex plane. It's important to understand the real part as it interacts with its imaginary counterpart to provide the full geometric positioning of a complex number.