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Chapter 7: 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

Identify the real and imaginary parts of each complex number. For the first complex number \((7+15i)\), the real part is 7 and the imaginary part is 15i. For the second complex number \((-11+14i)\), the real part is -11 and the imaginary part is 14i.
02

Add the Real Parts

Add the real parts of the complex numbers: \(7 + (-11) = -4\).
03

Add the Imaginary Parts

Add the imaginary parts of the complex numbers: \(15i + 14i = 29i\).
04

Combine the Results

Combine the results from step 2 and step 3 to get the final answer: \(-4 + 29i\).

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Key Concepts

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

Complex Number
A complex number is a number that includes both a real and an imaginary part. Think of it as a way to extend our understanding of numbers beyond the real number line.
  • They are usually written in the form \(a + bi\), where \a\ (the real part) and \b\ (the coefficient of the imaginary part) are real numbers, and \i\ is the imaginary unit with the property that \(i^2 = -1\).
  • This form combines a traditional real number part with a purely imaginary number part.
In the provided exercise, \(7 + 15i\) and \(-11 + 14i\) are examples of complex numbers. Each has a real part and an imaginary part, which brings us to our next key concepts.
Real Part
The real part of a complex number is simply the real number component. For a complex number in the form \(a + bi\), \a\ is the real part.
  • In the example \(7 + 15i\), the real part is 7.
  • For \(-11 + 14i\), the real part is -11.
Understanding the real part (and separating it from the imaginary part) is crucial when performing operations like addition or subtraction with complex numbers.
We first identify and then process these real parts separately, as demonstrated in Step 2 of the original exercise.
Imaginary Part
The imaginary part of a complex number refers to the component that involves the imaginary unit, \(i\). For a complex number in the form \(a + bi\), \bi\ is the imaginary part.
  • In the number \(7 + 15i\), the imaginary part is 15i.
  • In \(-11 + 14i\), the imaginary part is 14i.
It's important to remember that \(i\) signifies the square root of -1, adding a unique dimension to these numbers. When adding or subtracting complex numbers, we handle the imaginary parts separately, just as shown in Step 3 of the original solution.
Standard Form
The standard form of a complex number is \(a + bi\), where \a\ and \b\ are real numbers. This form is useful because it clearly separates the real part from the imaginary part.
  • When complex numbers are in this form, it's easier to see and work with their components.
  • For instance, after performing the operations in the exercise \(7 + 15i + (-11 + 14i)\), we find the result to be \(-4 + 29i\).
This result is in standard form, showing a clear real part (-4) and an imaginary part (29i). Keeping complex numbers in standard form ensures clarity and consistency in mathematical operations.

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