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Chapter 3: Problem 6

Find the real and imaginary parts of the complex number. $$-\frac{1}{2}$$

Short Answer

Expert verified
Real part: \( -\frac{1}{2} \), Imaginary part: 0.

Step by step solution

01

Identify the form of the complex number

A complex number is generally in the form of \( a + bi \), where \( a \) is the real part and \( b \) is the coefficient of the imaginary part \( i \).
02

Separate real and imaginary components

The given number is \( -\frac{1}{2} \). Here, it is expressed in the form \( a + bi \), where \( a = -\frac{1}{2} \) and \( b = 0 \). There is no imaginary part since the given number is purely real.

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Key Concepts

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

Real Part
Complex numbers can be intriguing to learn about, and understanding their structure is key. The real part of a complex number is denoted by the symbol "\( a \)" in the standard form \( a + bi \). Although complex numbers often feature both real and imaginary parts, not all of them do. For example, a complex number \(-\frac{1}{2}\) has only a real part.
This means that:
  • \( a = -\frac{1}{2} \)
  • There is no imaginary part (\
Imaginary Part
The imaginary part of a complex number is represented by \( b \) in the standard form \( a + bi \). Imaginary numbers are fascinating as they allow for expansions beyond traditional real number solutions. However, not all complex numbers have an imaginary component.
For the given example \(-\frac{1}{2}\):
  • \( b = 0 \) because there is no imaginary part (no "\(i\)" attached).
  • This makes the number purely real.
This shows that complex numbers are quite versatile since they can represent pure real numbers too. Understanding when the imaginary part is zero helps in recognizing real numbers within complex form.
Form of a Complex Number
Complex numbers follow a simple structure called the standard form, written as \( a + bi \). Here, "\( a\)" refers to the real part, and "\( b \times i\)" represents the imaginary part. This form helps us to categorize numbers as purely real, purely imaginary, or fully complex.
Let's break it down:
  • If \( a eq 0 \) and \( b = 0 \), the number is purely real.
  • If \( a = 0 \) and \( b eq 0 \), it is purely imaginary.
  • If both \( a eq 0 \) and \( b eq 0 \), the number is fully complex.
It's also important to remember that if \( b = 0 \), the term \( bi \) disappears, leaving us with a real number.
Understanding this structural form simplifies working with complex numbers, and aids in grasping their various types. In the given exercise, \(-\frac{1}{2}\) fits into this form with only the real number component "\( a \)."

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Most popular questions from this chapter

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=4 x^{4}-21 x^{2}+5$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{18}{(x-3)^{2}}$$

Show that the equation $$x^{5}-x^{4}-x^{3}-5 x^{2}-12 x-6=0$$ has exactly one rational root, and then prove that it must have either two or four irrational roots.

Use transformations of the graph of \(y=\frac{1}{x}\) to graph the rational function, as in Example 2. $$r(x)=\frac{1}{x+4}$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6$$
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