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Chapter 1: Problem 37

Determine the real and imaginary parts of the complex number. $$19 i$$

Short Answer

Expert verified
Real part: 0, Imaginary part: 19.

Step by step solution

01

Understanding the complex number

Complex numbers have the form: \( a + bi \) where \( a \) is the real part and \( bi \) is the imaginary part.
02

Identify the real part

For the given complex number \( 19i \), the real part is the coefficient of the imaginary unit \( i \). Since there is no separate term without \( i \), the real part is 0.
03

Identify the imaginary part

For the given complex number \( 19i \), the imaginary part is the coefficient of \( i \). Thus, the imaginary part is 19.
04

State the real and imaginary parts

The real part is 0 and the imaginary part is 19.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Part
When working with complex numbers, understanding the real part is essential. A complex number is typically written as: \( a + bi \). Here, \( a \) is the real part. It is the component of the complex number that does not involve the imaginary unit \( i \).
For example, in the complex number \( 3 + 4i \), the real part is 3.
In our exercise, the number is \( 19i \). Since there is no separate term that does not involve \( i \), the real part is 0.
Key points to remember about real parts of complex numbers:
  • The real part is the term without the \( i \) in a complex number.
  • It can be any real number, including zero.
Imaginary Part
Another critical component of complex numbers is the imaginary part. In the complex number form \( a + bi \), \( bi \) represents the imaginary part. Here, \( b \) is the coefficient that multiplies the imaginary unit \( i \).
For instance, envision the complex number \( 3 + 4i \). The imaginary part is \( 4i \), where 4 is the coefficient of \( i \).
Referring to our exercise, the number \( 19i \) only has the term \( 19i \) and no real part. Therefore, the imaginary part is simply 19.
Important points about the imaginary part include:
  • The imaginary part consists of any real number multiplied by \( i \).
  • It may be positive, negative, or zero.
Identifying Complex Numbers
Identifying complex numbers involves discerning both their real and imaginary parts. A complex number always follows the format \( a + bi \). Recognizing this form helps in breaking down the number into real and imaginary parts.
Let's revisit our exercise, \( 19i \). This number might appear confusing at first. However, knowing the standard format, we identify:
  • The real part is 0 (as there is no number without \( i \)).
  • The imaginary part is 19 (the coefficient of \( i \)).
To summarize, the main steps to identify complex numbers are:
  • Identify the term without \( i \) as the real part.
  • Identify the term with \( i \) as the imaginary part.
Understanding these steps makes working with complex numbers much more manageable.

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Most popular questions from this chapter

Determine the set of values for \(x\) for which the radical expression would produce a real number. For example, the expression \(\sqrt{x-1}\) is a real number if \(x-1 \geq 0\) or equivalently, \(x \geq 1\). a. \(\sqrt{2 x-9}\) b. \(\sqrt[4]{2 x-9}\)

Solve the equations. \(|k-3|=|k+3|\)

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Solve the inequality, and write the solution set in interval notation. \(\left|\frac{1}{4} q-2\right| \geq 0.05\)

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