Problem 7 - 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 part

A complex number is typically written in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. In the given complex number \( 5 - 7i \), identify which part corresponds to the real part.

Step 2: Extract the real part

From the complex number \( 5 - 7i \), notice that \( 5 \) is the real part and does not involve the imaginary unit \( i \). Therefore, the real part is \( 5 \).

Step 3: Identify the imaginary part

The imaginary part of a complex number involves the imaginary unit \( i \). In the expression \( 5 - 7i \), identify the term that includes \( i \).

Step 4: Extract the imaginary part

In \( 5 - 7i \), the term \(-7i \) represents the imaginary part. Hence, the coefficient \(-7\) is the imaginary component of this complex number.

Key Concepts

Real Part

In the world of complex numbers, the **real part** refers to the non-imaginary component of the number. It is the section without the imaginary unit 'i'. Complex numbers are usually written in the form \( a + bi \), where \( a \) represents the real part.

For instance, if we look at the complex number \( 5 - 7i \), the real part is directly visible as \( 5 \). It is the value without any imaginary component or 'i'.

• Real part: the component without 'i'
• Located before the 'i' term in the complex number

Understanding the real part is crucial because it helps us separate the complex number into components that can be mathematically manipulated more easily, such as in addition or multiplication of complex numbers.

Imaginary Part

The **imaginary part** of a complex number utilizes what is known as the imaginary unit, denoted by 'i'. This unit satisfies the equation \( i^2 = -1 \). In the complex number notation \( a + bi \), the portion \( bi \) indicates the imaginary part.

For the complex number \( 5 - 7i \), the imaginary part is represented by \(-7i\). Here, \(-7\) is the coefficient of the imaginary unit 'i'. Recognizing the imaginary portion is essential, as it carries all the properties that distinguish complex numbers from ordinary real numbers.

• Imaginary part includes and is associated with 'i'.
• Typically associated with equations that transcend standard algebra due to \( i^2 = -1 \).

Grasping the concept of the imaginary part allows us to resolve equations involving square roots of negative numbers, which don't exist in the realm of real numbers.

Complex Number Notation

The **complex number notation** is a vital mathematical framework that bridges real and imaginary numbers into one cohesive system. In its standard format \( a + bi \), a complex number employs two distinct parts: the real part and the imaginary part.

• \( a \) is the real part
• \( bi \) is the imaginary part

This notation allows mathematicians and engineers to work with numbers that describe real-world phenomena like alternating current in electrical engineering or wave functions in quantum physics. In our example, the complex number \( 5 - 7i \) shows precisely how these components are visually and functionally integrated.

Proper comprehension of this notation is crucial for effective manipulation and application of complex numbers in various mathematical operations and real-life applications. The dual nature of complex numbers simplifies many otherwise difficult problems by representing them in this straightforward yet powerful format.