Problem 12 - 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: -1/2; Imaginary part: 0.

Step 1: Identify the Complex Number

The problem states the number as \(-\frac{1}{2}\). First, recognize that complex numbers are generally of the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Step 2: Separate Real and Imaginary Parts

For the number \(-\frac{1}{2}\), note that it can be expressed as \(-\frac{1}{2} + 0i\) to fit the standard complex number form \(a + bi\). In this form, \(a = -\frac{1}{2}\) and \(b = 0\). Thus, the real part is \(-\frac{1}{2}\) and the imaginary part is \(0\).

Key Concepts

Real Part

The real part of a complex number represents the component that does not involve the imaginary unit, often denoted by the letter 'i'. In the standard form of a complex number, written as \(a + bi\), the real part is the number \(a\).
The real part is what you can plot along the horizontal axis in a complex plane, which helps visualize complex numbers by separating their real and imaginary components.
Understanding the real part can be straightforward once you recognize it as the integer or fractional part of the complex number that appears without an 'i'.

• In our example, the real part of the complex number \(-\frac{1}{2} + 0i\) is \(-\frac{1}{2}\).
• It's the value you would use if you were to describe the number on a regular number line, emphasizing the absence of any 'i' factor.

Imaginary Part

The imaginary part of a complex number involves multiplication by the imaginary unit \(i\), where \(i\) is defined such that \(i^2 = -1\). In the form \(a + bi\), the imaginary part is \(b\), which can be any real number.
This part determines the complex number's vertical position in the complex plane. The imaginary component essentially represents 'movement' or 'displacement' away from the real number line, giving full expression to two-dimensional vectors or points.
In practical terms, determining the imaginary part involves recognizing and isolating the coefficient of \(i\) from the complex number's expression.

• In your example, the imaginary part for \(-\frac{1}{2} + 0i\) is zero.
• It means no actual "imaginary" contribution is there, keeping the number purely real.

Complex Number Form

Complex numbers reside in a more intricate structure encapsulated by the form \(a + bi\). This standard form combines both the real and imaginary components to fully describe any number on the complex plane.
This dual nature makes complex numbers far more versatile than the numbers we're familiar with from the real number line, allowing mathematicians and engineers to handle scenarios where oscillations or rotations need precise tracking.
When sorting any number into this form, it becomes simple to deal with operations like addition or subtraction of complex numbers, by separately acting on their respective real and imaginary parts.

• For instance, identifying a number such as \(-\frac{1}{2}\) in the complex number form, you would write it as \(-\frac{1}{2} + 0i\).
• This expresses the number as 100% real with no imaginary part.

Recognizing how to transform regular numbers into this complex structure enhances problem-solving flexibility, whether dealing with electrical engineering concepts or advanced math problems.