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Chapter 3: Problem 11

Find the real and imaginary parts of the complex number. $$-\frac{2}{3} i$$

Short Answer

Expert verified
Real part: 0, Imaginary part: -2/3.

Step by step solution

01

Understand the Format of a Complex Number

A complex number is usually expressed in the form \( a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part.
02

Identify the Given Complex Number

The complex number given in the problem is \( -\frac{2}{3} i \). This is already in the form of \( a + bi \), where \( a = 0 \) and \( b = -\frac{2}{3} \).
03

Extract the Real Part

Identify the real part of the complex number. Since the complex number is \( -\frac{2}{3} i \), there is no real number added to the imaginary part \( bi \). Thus, the real part \( a \) is \( 0 \).
04

Extract the Imaginary Part

Identify the imaginary part of the complex number. In \( -\frac{2}{3} i \), the imaginary part \( b \) is \( -\frac{2}{3} \), which multiplies the imaginary unit \( i \).

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Key Concepts

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

Real Part of Complex Numbers
A complex number consists of two parts: the real part and the imaginary part. The real part of a complex number is any number without the imaginary unit, represented by \( i \).
The standard format of a complex number is \( a + bi \), where \( a \) is the real part. In many cases, the real part is immediately obvious, such as in the number 3 + 4i, where 3 is the real part.
However, sometimes only the imaginary part is present, as in the original exercise, where the number is \(-\frac{2}{3} i\). Here, the real part is simply 0 because there is no number before the imaginary part or the imaginary unit.
Imaginary Part of Complex Numbers
The imaginary part of a complex number involves the coefficient that multiplies the imaginary unit \( i \). It adds a new dimension to numbers, allowing for the representation of numbers that cannot appear on the traditional number line.
Written in the form \( a + bi \), the imaginary part is \( b \). In the complex number \(-\frac{2}{3} i\), the imaginary part is \(-\frac{2}{3}\). This means that the entire value of the complex number is concentrated in the imaginary dimension, with no real component present.
  • In expressions like \(-\frac{2}{3} i\), it's crucial to recognize that \( b \) is \(-\frac{2}{3}\).
  • The negative sign indicates the direction on the imaginary axis.
Imaginary Unit 'i'
The imaginary unit \( i \) is fundamental in defining complex numbers. By convention, \( i \) is defined as the square root of \(-1\), represented as \( i^2 = -1 \).
This property allows for the extension of real numbers to complex numbers, providing solutions to equations like \( x^2 + 1 = 0 \), which have no solution in the realm of real numbers alone.
  • The imaginary unit \( i \) plays an essential role in distinguishing imaginary parts from real parts.
  • When you multiply a real number by \( i \), it transitions the number into the imaginary dimension.
For example, in the expression \(-\frac{2}{3} i\), \(-\frac{2}{3}\) represents the coefficient, while \( i \) places this quantity in the imaginary realm, far from our everyday real numbers.

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Most popular questions from this chapter

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

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, comect to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{5}}{x^{3}-1}$$

Graph the rational function \(f\), and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, \quad g(x)=1-x^{2}$$

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. \(z=\bar{z}\) if and only if \(z\) is real.

Evaluate the radical expression and express the result in the form \(a+b i\) $$\frac{\sqrt{-36}}{\sqrt{-2} \sqrt{-9}}$$
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