Problem 10 - Precalculus Mathematics for Calculus

Question

Find the real and imaginary parts of the complex number. $$-\frac{1}{2}$$

Step-by-Step Solution

Answer

Real part: \(-\frac{1}{2}\), Imaginary part: 0.

Step 1: Identify the Form of the Complex Number

A complex number is generally written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

Step 2: Recognize the Given Number

The given number is \(-\frac{1}{2}\). This number can be expressed as \(-\frac{1}{2} + 0i\).

Step 3: Determine the Real Part

In the expression \(-\frac{1}{2} + 0i\), the real part \( a \) is \(-\frac{1}{2}\).

Step 4: Determine the Imaginary Part

In the same expression \(-\frac{1}{2} + 0i\), the imaginary part \( bi \) is \(0 \cdot i\), thus the imaginary part is \(0\).

Key Concepts

Real Part

When dealing with complex numbers, understanding the real part is crucial. A complex number is typically written as \( a + bi \), where \( a \) represents the real part. In essence, the real part is simply the component of the complex number that does not involve the imaginary unit \( i \).
Consider the number \(-\frac{1}{2} + 0i\). The real part here is \(-\frac{1}{2}\).

• The real part provides you with an insight into the number's position on the real axis, which runs horizontally in the complex plane.
• Notably, if there is no explicit imaginary component, it simply means the imaginary part is zero. However, the real part can exist independently as in our example, \(-\frac{1}{2}\).

Imaginary Part

Next, let's explore the imaginary part of a complex number. In the form \( a + bi \), the imaginary part is the term \( bi \) where \( b \) is a real number and \( i \) is the imaginary unit, which is defined by the property \( i^2 = -1 \).
In our specific example, \(-\frac{1}{2} + 0i\), the imaginary component is \(0 \cdot i\).

• This means the imaginary part in this case is \(0\). When a complex number's imaginary part is zero, the number essentially lies on the real axis.
• Even so, it's standard practice to write the complex number in the form \( a + bi \), as this helps visualize it in both the real and imaginary dimensions, enhancing understanding.

Complex Number Form

To tie everything together, let's talk about the form of a complex number, always written as \( a + bi \). This notation comprises both the real and imaginary parts:

• \( a \) is the real part, situated along the real axis in the complex plane.
• \( bi \) is the imaginary part, aligned perpendicularly along the imaginary axis.

As presented earlier with the example \(-\frac{1}{2} + 0i\):

• The real part \( a \) is \(-\frac{1}{2}\).
• The imaginary part \( bi \) is \(0\), simplifying to a real number with no vertical displacement on the imaginary axis.

This form allows mathematicians and students alike to clearly grasp not just numerical value, but also geometric implications in the complex plane. Even when one part is zero, expressing the number in this form can be very enlightening. Understanding both components and their interactions is foundational to mastering complex numbers.