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Chapter 7: Problem 48

Add or subtract as indicated. Give answers in standard form. $$(-1+i)+(2+5 i)+(3+2 i)$$

Short Answer

Expert verified
4 + 8i

Step by step solution

01

Group Real Parts Together

Identify and group all the real parts from each complex number. Here, the real parts are -1, 2, and 3.
02

Group Imaginary Parts Together

Next, identify and group all the imaginary parts from each complex number. The imaginary parts are 1, 5, and 2.
03

Add Real Parts

Add the real parts together: -1 + 2 + 3 = 4.
04

Add Imaginary Parts

Add the imaginary parts together: 1 + 5 + 2 = 8. Therefore, the total imaginary part is 8i.
05

Form the Standard Form

Combine the results from the steps above to form the final complex number in standard form: 4 + 8i.

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Key Concepts

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

adding complex numbers
Adding complex numbers is easier than you might think.
Complex numbers are numbers that have a real part and an imaginary part.
To add them, you just add their real parts together and their imaginary parts together.
For example, in the exercise \((-1+i)+(2+5i)+(3+2i)\), we first group the real and imaginary parts separately.
  • Real parts: -1, 2, 3
  • Imaginary parts: 1, 5, 2
By adding the real parts \((-1+2+3)\), we get 4. Next, add the imaginary parts \((1+5+2)\), which results in 8i.
The combined result is \(4+8i\). Simple, right?
standard form of complex numbers
The standard form of a complex number is very easy to remember.
A complex number is usually written in the form \(a+bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
In our example \((-1+i)+(2+5i)+(3+2i)\), after adding the numbers, we get \(4+8i\). Notice how 4 is the real part and 8i is the imaginary part.
Having the complex number in standard form \(a+bi\) makes calculations easier and the results clearer.
real and imaginary parts
Understanding the real and imaginary parts of a complex number is crucial.
A complex number combines both a real part and an imaginary part.
The real part is the number without the \(i\), and the imaginary part is the number with the \(i\).
For example, in the number \(4+8i, 4\) is the real part and \(8i\) is the imaginary part.
When performing operations with complex numbers, always separate and handle the real and imaginary parts separately.
This way, you can clearly see how they contribute to the final number.

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Most popular questions from this chapter

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