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Chapter 3: Problem 10

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 written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.
02

Recognize the Given Number

The given number is \(-\frac{1}{2}\). This number can be expressed as \(-\frac{1}{2} + 0i\).
03

Determine the Real Part

In the expression \(-\frac{1}{2} + 0i\), the real part \( a \) is \(-\frac{1}{2}\).
04

Determine the Imaginary Part

In the same expression \(-\frac{1}{2} + 0i\), the imaginary part \( bi \) is \(0 \cdot i\), thus the imaginary part is \(0\).

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Key Concepts

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

Real Part
When dealing with complex numbers, understanding the real part is crucial. A complex number is typically written as \( a + bi \), where \( a \) represents the real part. In essence, the real part is simply the component of the complex number that does not involve the imaginary unit \( i \).
Consider the number \(-\frac{1}{2} + 0i\). The real part here is \(-\frac{1}{2}\).
  • The real part provides you with an insight into the number's position on the real axis, which runs horizontally in the complex plane.
  • Notably, if there is no explicit imaginary component, it simply means the imaginary part is zero. However, the real part can exist independently as in our example, \(-\frac{1}{2}\).
Imaginary Part
Next, let's explore the imaginary part of a complex number. In the form \( a + bi \), the imaginary part is the term \( bi \) where \( b \) is a real number and \( i \) is the imaginary unit, which is defined by the property \( i^2 = -1 \).
In our specific example, \(-\frac{1}{2} + 0i\), the imaginary component is \(0 \cdot i\).
  • This means the imaginary part in this case is \(0\). When a complex number's imaginary part is zero, the number essentially lies on the real axis.
  • Even so, it's standard practice to write the complex number in the form \( a + bi \), as this helps visualize it in both the real and imaginary dimensions, enhancing understanding.
Complex Number Form
To tie everything together, let's talk about the form of a complex number, always written as \( a + bi \). This notation comprises both the real and imaginary parts:
  • \( a \) is the real part, situated along the real axis in the complex plane.
  • \( bi \) is the imaginary part, aligned perpendicularly along the imaginary axis.
As presented earlier with the example \(-\frac{1}{2} + 0i\):
  • The real part \( a \) is \(-\frac{1}{2}\).
  • The imaginary part \( bi \) is \(0\), simplifying to a real number with no vertical displacement on the imaginary axis.
This form allows mathematicians and students alike to clearly grasp not just numerical value, but also geometric implications in the complex plane. Even when one part is zero, expressing the number in this form can be very enlightening. Understanding both components and their interactions is foundational to mastering complex numbers.

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Most popular questions from this chapter

A drug is administered to a patient, and the concentration of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in hours since giving the drug), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$ c(t)=\frac{5 t}{t^{2}+1} $$ Graph the function \(c\) with a graphing device. (a) What is the highest concentration of drug that is reached in the patient's bloodstream? (b) What happens to the drug concentration after a long period of time? (c) How long does it take for the concentration to drop below \(0.3 \mathrm{mg} / \mathrm{L} ?\)

After a certain drug is injected into a patient, the concentration \(c\) of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in minutes since the injection), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$ c(t)=\frac{30 t}{t^{2}+2} $$ (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of drug in the bloodstream?

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z \cdot \bar{z}\) is a real number.

Snow began falling at noon on Sunday. The amount of snow on the ground at a certain location at time \(t\) was given by the function $$ \begin{aligned} h(t)=11.60 t &-12.41 t^{2}+6.20 t^{3} \\ &-1.58 t^{4}+0.20 t^{5}-0.01 t^{6}\end{aligned}$$ where \(t\) is measured in days from the start of the snowfall and \(h(t)\) is the depth of snow in inches. Draw a graph of this function, and use your graph to answer the following questions. (a) What happened shortly after noon on Tuesday? (b) Was there ever more than 5 in. of snow on the ground? If so, on what day(s)? (c) On what day and at what time (to the nearest hour) did the snow disappear completely?

As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the observer than it would if the train were at rest, because the crests of the sound waves are compressed closer together. This phenomenon is called the Doppler effect. The observed pitch \(P\) is a function of the speed \(v\) of the train and is given by $$ P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right) $$ where \(P_{0}\) is the actual pitch of the whistle at the source and \(s_{0}=332 \mathrm{m} / \mathrm{s}\) is the speed of sound in air. Suppose that a train has a whistle pitched at \(P_{n}=440 \mathrm{Hz}\). Graph the function \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically? (IMAGES CANNOT COPY)
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