Problem 8 - Precalculus: Mathematics for Calculus

Question

Find the real and imaginary parts of the complex number. $$-6+4 i$$

Step-by-Step Solution

Answer

The real part is -6, and the imaginary part is 4.

Step 1: Identify the Complex Number

The complex number is given as \(-6 + 4i\). In general, a complex number is expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Step 2: Determine the Real Part

To find the real part of the complex number, look at the term without the imaginary unit \(i\). In \(-6 + 4i\), this term is \(-6\). Thus, the real part is \(-6\).

Step 3: Determine the Imaginary Part

To find the imaginary part, look at the coefficient of the imaginary unit \(i\). In \(-6 + 4i\), the coefficient is \(4\). Therefore, the imaginary part of the complex number is \(4\).

Key Concepts

Real Part of Complex Numbers

The real part of a complex number is a crucial component that defines how the number behaves in real-number scenarios. When examining any complex number, which is generally written as \(a + bi\), you'll find that the real part is the "\(a\)" portion. This part consists of the terms that do not involve the imaginary unit \(i\). For instance, if we consider the complex number \(-6 + 4i\), the real part is \(-6\). This means that on a number line, the real component aligns with the regular real numbers you often use in arithmetic.

In summary, here's how to handle the real part:

• Identify the terms without the \(i\).
• The real part is simply the coefficient that stands alone in real form.

This allows you to easily extract the real value from the complex number, making further calculations more manageable.

Imaginary Part of Complex Numbers

Understanding the imaginary part of a complex number helps you dive deeper into the rich world beyond real numbers. The imaginary piece is represented in the general form \(bi\), where \(b\) is a real number called the coefficient of the imaginary unit \(i\). To find this part in a complex number like \(-6 + 4i\), you simply need to locate the term that includes the \(i\). Here, the imaginary part is \(4\) because it is the multiplier of \(i\).

Identifying the imaginary component can be summarized like this:

• Look for the term accompanied by \(i\).
• The number in front of \(i\) is the imaginary part.

Though "imaginary," this part plays a vital role in calculations, especially in fields like engineering and physics, where complex numbers provide valuable insights.

Complex Number Representation

Complex numbers provide a way to solve equations and represent numbers that extends beyond the real number line. They are written as \(a + bi\) where \(a\) is the real part, and \(bi\) is the imaginary part. This combination allows for a unique representation on a two-dimensional plane known as the complex plane, where the horizontal axis is for the real part and the vertical axis is for the imaginary part.

Some key pointers for understanding complex number representation include:

• Each complex number corresponds to a point on the complex plane.
• The coordinate \((a, b)\) reflects the real and imaginary parts, respectively.

The complex plane helps in visualizing complex numbers, making it easier to perform operations like addition, subtraction, multiplication, and division. Through this visual representation, complex numbers become not just an abstract concept but a more tangible tool in mathematics and applied sciences.