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Chapter 2: Problem 24

Perform the operation and write the result in standard form.$$(13-2 i)+(-5+6 i)$$

Short Answer

Expert verified
The result of the operation is \(8 + 4i\).

Step by step solution

01

Identify the Real and Imaginary Components

In the given complex numbers, for \(13-2i\), 13 is the real part and \(-2i\) is the imaginary part. Similarly, for \(-5+6i\), \(-5\) is the real part and \(6i\) is the imaginary part.
02

Add the Real Components

Add the real components of both complex numbers, which are 13 and -5. So, \(13 + (-5) = 8\)
03

Add the Imaginary Components

Next, add the imaginary components of both complex numbers, which are \(-2i\) and \(6i\). So, \(-2i + 6i = 4i\)
04

Write the Result in Standard Form

Combine the results from step 2 and step 3 to give the final answer. So, the sum of the two complex numbers is \(8 + 4i\), which is the standard form of a complex number.

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Key Concepts

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

Real Components
In the world of complex numbers, each number is composed of two parts: the real component and the imaginary component. Identifying these components is the first step to understanding any complex number.
A real component is the part of a complex number that doesn't involve the imaginary unit. For example, in the complex number \(13 - 2i\), the real part is 13. Similarly, in \(-5 + 6i\), the real part is -5. This real component behaves just like any other number on the number line.
Remember that any complex number \(a + bi\) has 'a' as its real component, where 'a' is a real number.
Imaginary Components
Imaginary components of complex numbers include the imaginary unit 'i', which is defined as \(i = \sqrt{-1}\). Imaginary components allow for mathematical operations that aren鈥檛 possible with real numbers alone.
For example, in the complex number \(13 - 2i\), \(-2i\) is the imaginary part, and in the number \(-5 + 6i\), \(6i\) is the imaginary part. 'i' essentially represents the \'vertical\' direction in the complex plane, which adds a new dimension to our calculations.
Standard Form
Complex numbers are typically expressed in what is known as the standard form. This form neatly arranges the real and imaginary components as \(a + bi\).
The standard form makes it easy to perform arithmetic operations and to visualize complex numbers on the complex plane. For addition and subtraction, the numbers can be combined directly by their respective components. In the example given, \(8 + 4i\) is in standard form: 8 is the real part and 4\(i\) represents the imaginary component.
Addition of Complex Numbers
Adding complex numbers is straightforward once you identify and separate the real and imaginary components. The real parts are added together, and the imaginary parts are added separately.
In our example, first add the real components: \(13 + (-5) = 8\). Then, add the imaginary components: \(-2i + 6i = 4i\).
Finally, combining these results, we obtain \(8 + 4i\). Adding complex numbers in this way ensures every part is accounted for and keeps the sum in standard form.

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Most popular questions from this chapter

Graphical Analysis From 1950 through 2005 , the per capita consumption \(C\) of cigarettes by Americans \(\begin{array}{lllllll}\text { (age } & 18 & \text { and } & \text { older) } & \text { can } & \text { be } & \text { modeled } & \text { by }\end{array}\) \(C=3565.0+60.30 t-1.783 t^{2}, 0 \leq t \leq 55,\) where \(t\) is the year, with \(t=0\) corresponding to 1950 . (Source: Tobacco Outlook Report) (a) Use a graphing utility to graph the model. (b) Use the graph of the model to approximate the maximum average annual consumption. Beginning in \(1966,\) all cigarette packages were required by law to carry a health warning. Do you think the warning had any effect? Explain. (c) In 2005, the U.S. population (age 18 and over) was \(296,329,000 .\) Of those, about 59,858,458 were smokers. What was the average annual cigarette consumption per smoker in 2005? What was the average daily cigarette consumption per smoker?

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$h(x)=4 x^{2}-4 x+21$$

Use a graphing utility to graph the quadratic function. Find the \(x\) -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when \(f(x)=0\) $$f(x)=2 x^{2}-7 x-30$$

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=x^{2}-6 x$$

Find all real zeros of the function. $$f(y)=4 y^{3}+3 y^{2}+8 y+6$$
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