Problem 6 - Algebra and Trigonometry

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 General Form of a Complex Number

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

Step 2: Write the Given Number in Complex Form

The given number is \(-\frac{1}{2}\), which can be written in the complex form as \(-\frac{1}{2} + 0i\). Here, \( a = -\frac{1}{2} \) and \( b = 0 \).

Step 3: Extract the Real Part

From the complex number \(-\frac{1}{2} + 0i\), the real part is the coefficient of the real term, which is \( -\frac{1}{2} \).

Step 4: Extract the Imaginary Part

From the complex number \(-\frac{1}{2} + 0i\), the imaginary part is the coefficient of the imaginary term \( i \), which is \( 0 \).

Key Concepts

Real Part

The real part of a complex number is the number that appears in the position of "a" in the expression \(a + bi\). It is simply a pure number without any imaginary unit "i" attached to it. In the context of the example provided, the given complex number is \(-\frac{1}{2} + 0i\). Thus, the real part is \(-\frac{1}{2}\).
This is because it is the number that directly relates to the real axis on the complex plane.
Understanding the real part is crucial since it represents the actual numeric value without the imaginary component.

• A complex number can have a negative real part, as in this case.
• This real part affects the horizontal direction of the number on the complex plane.
• In calculations, the real part is handled just like any other real number.

Imaginary Part

The imaginary part of a complex number is the number in front of the imaginary unit "i". In the expression \(a + bi\), the "b" represents the imaginary part. For the number \(-\frac{1}{2} + 0i\), the imaginary part is 0.
This means that there is no imaginary component in this particular number.

• The imaginary part tells us how much the number extends in the vertical direction on the complex plane.
• If the imaginary part is zero, the complex number lies entirely on the real axis.
• Imaginary numbers are rooted in mathematical operations such as the square root of negative numbers.

When working with complex numbers, identifying the imaginary part helps understand the full scope of the number's position in the complex number system.

Complex Form

Complex form denotes a standard way to write complex numbers as \(a + bi\). It combines both real and imaginary parts into a single expression. In our example, \(-\frac{1}{2}\) is initially considered a real number, but when expressed as a complex number, it takes the form \(-\frac{1}{2} + 0i\). Thus, it adheres to the conventional form, combining both components even if the imaginary part is zero.

Expressing numbers in complex form is beneficial in numerous mathematical applications, including:

• Simplifying calculations involving square roots of negative numbers.
• Analyzing electrical circuits and control systems in engineering.
• Calculating oscillations and waves in physics.

Understanding complex form is essential as it lays the foundation for advanced mathematical concepts like complex functions and transformations.