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Chapter 3: Problem 95

Divide as indicated. Write each quotient in standand form. $$\frac{9+2 i}{2+i}$$

Short Answer

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

Step by step solution

01

Understand the Problem

We need to divide the complex number \(9 + 2i\) by \(2 + i\) and express the result in standard form \(a + bi\).
02

Multiply by the Conjugate

To simplify this division, multiply the numerator and the denominator by the conjugate of the denominator \(2 - i\):\[\frac{9 + 2i}{2 + i} \cdot \frac{2 - i}{2 - i}\]
03

Simplify the Denominator

Using the identity \((a+b)(a-b) = a^2 - b^2\), calculate the denominator:\[(2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 5\]
04

Distribute in the Numerator

Now, distribute \(9 + 2i\) with \(2 - i\):\[(9 + 2i)(2 - i) = 9 \cdot 2 + 9 \cdot (-i) + 2i \cdot 2 + 2i \cdot (-i)\] Solve the individual terms to get:\[18 - 9i + 4i - 2i^2 = 18 - 9i + 4i + 2\] Using \(i^2 = -1\), this simplifies to:\[18 + 2 - 5i = 20 - 5i\]
05

Write the Result in Standard Form

Divide each term of the result by the denominator 5:\[\frac{20}{5} - \frac{5i}{5} = 4 - i\] Thus, the division results in the complex number \(4 - i\).

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Key Concepts

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

Complex Division
Dividing complex numbers is a bit different from dividing real numbers, but it follows a structured process that makes the operation straightforward. The goal is to find the quotient of two complex numbers. Just like in regular math, division can be seen as multiplying by a reciprocal. However, with complex numbers, we need to use the conjugate to perform this task.

Here鈥檚 what you do:
  • Identify the numerator and the denominator of the complex division problem.
  • Multiply both the numerator and the denominator by the conjugate of the denominator.
  • This multiplication clears the imaginary part of the denominator, effectively transforming it into a real number.
By following these steps, you simplify the problem into one that is easier to manage, ultimately allowing the result to be expressed in standard form.
Standard Form
When we work with complex numbers, it is often helpful to express them in a standard form, which is given by the expression \(a + bi\). In this standard form, \(a\) is the real part, and \(b\) is the coefficient of the imaginary part \(i\).

Some key points about standard form:
  • It clearly separates the real and imaginary components of the number.
  • By maintaining this form, complex arithmetic remains consistent and understandable.
  • This representation allows complex numbers to be plotted on a two-dimensional plane, enhancing visual understanding.
Using standard form, division of complex numbers becomes a more approachable task. Once the computation is complete, it helps in easily identifying the real and imaginary parts of the quotient.
Conjugate
Conjugates play a crucial role in dividing complex numbers. The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of \(a + bi\) is \(a - bi\).

Here's why conjugates are important in complex division:
  • They allow us to eliminate the imaginary unit \(i\) from the denominator.
  • Multiplying a complex number by its conjugate results in a real number, specifically \(a^2 + b^2\).
  • This property is used to "clean" the denominator, which simplifies the division process.
Using conjugates not only simplifies the division but also ensures the end result fits neatly into the standard form, making it easier to interpret and work with.

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Most popular questions from this chapter

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x^{2}-9 x>-4\) (b) \(2 x^{2}-9 x \leq-4\)

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=2 x-5$$

Solve each problem. Selected values of the stopping distance \(y\) in feet of a car traveling \(x\) mph are given in the table. $$\begin{array}{|c|c|}\hline \begin{array}{c} \text { Speed } \\ \text { (in mph) } \end{array} & \begin{array}{c} \text { Stopping Distance } \\ \text { (in feet) } \end{array} \\ \hline 20 & 46 \\ 30 & 87 \\ 40 & 140 \\ 50 & 240 \\ 60 & 282 \\ 70 & 371 \\ \hline \end{array}$$ (a) Plot the data. (b) The quadratic function $$f(x)=0.056057 x^{2}+1.06657 x$$ is one model for the data. Find and interpret \(f(45)\) (c) Graph the function in the same window as the data to determine how well \(f\) models the stopping distance.

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x^{2}-x+3<0\) (b) \(2 x^{2}-x+3 \geq 0\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}+6 x+8<0\) (b) \(x^{2}+6 x+8 \geq 0\)
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