Problem 5 - Precalculus Mathematics for Calculus

Question

Find the real and imaginary parts of the complex number. $$5-7 i$$

Step-by-Step Solution

Answer

Real part: 5, Imaginary part: -7.

Step 1: Identify the Real Component

The real part of a complex number is the component that does not have the imaginary unit, \(i\), attached to it. In the complex number \(5 - 7i\), the real part is \(5\).

Step 2: Identify the Imaginary Component

The imaginary part of a complex number is the coefficient of the imaginary unit \(i\). In the complex number \(5 - 7i\), the imaginary part is \(-7\).

Key Concepts

Real Part

Every complex number is made up of a real part and an imaginary part. The real part of a complex number is simply the component that doesn’t involve the imaginary unit, denoted by the letter "i." When given a complex number such as \(5 - 7i\), the real part is quite straightforward to identify.
Simply look for the number that stands alone without any multiplication by the imaginary unit.
For \(5 - 7i\), the real part is:

• \(5\)

Identifying the real part is a simple yet crucial step in working with complex numbers as it clarifies the number's position on the real axis of the complex plane.
It's important to know that even though the real part may seem intuitive, it plays a significant role in understanding complex numbers and their interactions.

Imaginary Part

The imaginary part of a complex number involves the coefficient that is multiplied by the imaginary unit, represented as \(i\). This part is what makes complex numbers unique, as it doesn't correspond to any point on the traditional number line.
For the complex number \(5 - 7i\), the imaginary part can be found by:

• Identifying the coefficient in front of the \(i\), which is \(-7\).

The imaginary part \(-7\) means that if you were to plot this number on the complex plane, you would move downward by 7 units.
Remember that the imaginary part is always associated with the "i" and it's crucial for performing operations such as addition, subtraction, and even multiplication with other complex numbers.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a special mathematical symbol that represents the square root of \(-1\). This is not something we encounter in real numbers, making it a fascinating concept in mathematics, particularly in complex numbers.
The definition of \(i\) is core to understanding how complex numbers work and behave. In the expression \(5 - 7i\):

• The "\(i\)" signifies that \(-7\) is the imaginary part.
• It indicates a direction on the imaginary axis of the complex plane.

The imaginary unit allows us to work with equations that have no real solutions, such as \(x^2 + 1 = 0\).
Understanding \(i\) is crucial because it extends the idea of numbers beyond the one-dimensional real line to the two-dimensional complex plane.
This opens up a whole new world of mathematical exploration and application.