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Chapter 2: Problem 55

Write the quotient in standard form. \(\frac{9-4 i}{i}\)

Short Answer

Expert verified
The quotient in standard form is \(-4 - 9i\).

Step by step solution

01

Multiply by the conjugate

To remove \(i\) from the denominator, we multiply the numerator and denominator by the conjugate of \(i\), which is \(-i\), since the conjugate of \(a + bi\) is \(a - bi\), and in our case \(a = 0\) and \(b = 1\): \(\frac{9-4 i}{i} * \frac{-i}{-i}\)
02

Multiply the numerators

Now multiply the numerators: \((9 - 4i) * (-i) = -9i + 4i^2\). Remember \(i^2 = -1\), so we can simplify this to \(-9i - 4\)
03

Multiply the denominators

In this case, multiplying the denominators, we find \(i * -i = -i^2\). But since \(i^2 = -1\), this simplifies to 1.
04

Final Form

Dividing the numerator by the denominator, we get \(-9i - 4\), which in standard form is \(-4 - 9i\).

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Key Concepts

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

Standard Form of Complex Numbers
Complex numbers are expressed in the form of a real part and an imaginary part. This form is known as the standard form of complex numbers. The standard form is written as \(a + bi\), where \(a\) represents the real part and \(bi\) represents the imaginary part. Here are some helpful points to remember:

  • The real part \(a\) is any real number.
  • The imaginary part \(bi\) includes the imaginary unit \(i\), where \(b\) is also a real number.

When performing operations with complex numbers, it is important to express the final result in this standard form. In our case, the problem starts with a fraction, and we simplify it into the form \(-4 - 9i\), where \(-4\) is the real part and \(-9\) is the imaginary part.
Conjugates of Complex Numbers
The conjugate of a complex number is a key concept in complex numbers division.
The conjugate helps in removing imaginary and complex parts from the denominator.
For a complex number \(a + bi\), the conjugate is \(a - bi\).
This means you simply flip the sign of the imaginary part. When you multiply a complex number by its conjugate, you get a real number. This is because:

  • For example, \((a + bi) \times (a - bi) = a^2 - b^2i^2 = a^2 + b^2\) since \(i^2 = -1\).
  • In the division problem \(\frac{9-4i}{i}\), to eliminate \(i\) from the denominator, we used the conjugate of \(i\), which is \(-i\).

This allows the problem to be simplified into standard form.
Imaginary Unit Properties
The imaginary unit \(i\) is central to working with complex numbers.
It follows specific rules that are crucial in calculations.
The most important properties are:

  • The definition: \(i = \sqrt{-1}\). It's the number whose square is \(-1\).
  • Squaring \(i\) gives \(i^2 = -1\), a universal property used in complex number calculations.

These properties assist in simplifying expressions, especially in division. For instance, in the example \((9 - 4i) \times (-i)\), we use \(i^2 = -1\) to simplify \(4i^2\) into \(-4\).
Having a good grasp of these rules makes it easier to work with complex numbers in any form, whether in real-life problems or mathematical exercises.

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Most popular questions from this chapter

The mean salaries \(S\) (in thousands of dollars) of classroom teachers in the United States from 2000 through 2007 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, } S \\\\\hline 2000 & 42.2 \\ 2001 & 43.7 \\\2002 & 43.8 \\ 2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\\\hline\end{array}$$ A model that approximates these data is given by \(S=\frac{42.6-1.95 t}{1-0.06 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. (Source: Educational Research Service, Arlington, VA) (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) According to the model, in what year will the salary for classroom teachers exceed \(\$ 60,000 ?\) (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.

Solve the inequality and graph the solution on the real number line. \(\frac{x^{2}-1}{x}<0\)

Find the domain of \(x\) in the expression. Use a graphing utility to verify your result. \(\sqrt{x^{2}-9 x+20}\)

Solve the inequality and graph the solution on the real number line. \(\frac{x^{2}+2 x}{x^{2}-9} \leq 0\)

Write a rational function \(f\) that has the specified characteristics. (There are many correct answers.) (a) Vertical asymptote: \(x=2\) Horizontal asymptote: \(y=0\) Zero: \(x=1\) (b) Vertical asymptote: \(x=-1\) Horizontal asymptote: \(y=0\) Zero: \(x=2\) (c) Vertical asymptotes: \(x=-2, x=1\) Horizontal asymptote: \(y=2\) Zeros: \(x=3, x=-3\), (d) Vertical asymptotes: \(x=-1, x=2\) Horizontal asymptote: \(y=-2\) Zeros: \(x=-2, x=3\)
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