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Chapter 10: Problem 40

Add or subtract as indicated. Give answers in standard form. $$ (7+15 i)+(-11+14 i) $$

Short Answer

Expert verified
-4 + 29i

Step by step solution

01

Identify Real and Imaginary Parts

Write down the real parts and imaginary parts separately from the complex numbers. From the given numbers, the real parts are 7 and -11. The imaginary parts are 15i and 14i.
02

Add the Real Parts

Add the real parts together: (7) + (-11) = -4
03

Add the Imaginary Parts

Add the imaginary parts together: (15i) + (14i) = 29i
04

Combine the Results

Combine the sums of the real parts and the imaginary parts to form the final complex number: -4 + 29i

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Key Concepts

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

Real and Imaginary Parts
Complex numbers have two main components: the real part and the imaginary part. The real part is simply a regular number, while the imaginary part is paired with the imaginary unit, denoted as 'i'. For example, in the complex number \(7 + 15i\), 7 is the real part and 15i is the imaginary part.

Imaginary numbers are based on the square root of -1, which is represented by 'i'. This might sound confusing, but think of 'i' as just a tag that helps identify the imaginary part.

Commonly, a complex number is represented as \(a + bi\), where:
  • 'a' is the real part.
  • 'bi' is the imaginary part.
Always identify and separate these parts before performing any operations like addition or subtraction.
Addition of Complex Numbers
Adding complex numbers is straightforward. You simply add the real parts together and the imaginary parts together separately. Let's take the numbers from the exercise: \((7+15i)+(-11+14i)\).

Identify the real parts (7 and -11) and the imaginary parts (15i and 14i).
  • Add the real parts: \(7 + (-11) = -4\)
  • Add the imaginary parts: \(15i + 14i = 29i\)
Combine these results to get the final simplified form: \(-4 + 29i\).

This method keeps the real and imaginary components properly organized and ensures accurate results.
Standard Form of Complex Numbers
The standard form of a complex number is written as \(a + bi\). To ensure clarity and consistency, always present your complex numbers in this format.

In the exercise, we composed the result \(-4 + 29i\) in standard form. Here, -4 is the real part (a) and 29i is the imaginary part (bi).

Even when dealing with negative or zero values:
  • A zero real part: \(0 + 29i\) simplifies to \(29i\).
  • A zero imaginary part: \(-4 + 0i\) simplifies to \(-4\).
Writing in standard form helps in performing further operations and makes communication clearer.

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Most popular questions from this chapter

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A student rationalized the following denominator as shown. $$\frac{5}{\sqrt[3]{2}}=\frac{5 \cdot \sqrt[3]{2}}{\sqrt[3]{2} \cdot \sqrt[3]{2}}=\frac{5 \sqrt[3]{2}}{2}$$

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Simplify \(-\sqrt[4]{1250}\)

Work each problem. Replace \(a\) with 3 and \(b\) with 4 to show that, in general, $$ \sqrt{a^{2}+b^{2}} \neq a+b $$
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