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Chapter 7: Problem 29

Add or subtract and simplify. Write each answer in the form \(a+b i\). $$ (9+5 i)-(2+3 i) $$

Short Answer

Expert verified
7 + 2i

Step by step solution

01

Identify the Real and Imaginary Parts

Write down the real and imaginary parts of each complex number. For the first complex number, 9 is the real part and 5i is the imaginary part. For the second complex number, 2 is the real part and 3i is the imaginary part.
02

Subtract the Real Parts

Subtract the real parts of the complex numbers: \(9 - 2\). This equals 7.
03

Subtract the Imaginary Parts

Subtract the imaginary parts of the complex numbers: \(5i - 3i\). This equals 2i.
04

Combine the Results

Combine the results from the previous steps to write the answer in the form \(a+bi\). So, \(7 + 2i\).

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Key Concepts

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

real parts
In a complex number, the real part is the term that does not include the imaginary unit i. For example, in the complex number \(9 + 5i\), 9 is the real part.
Similarly, in the complex number \(2 + 3i\), 2 is the real part.
When subtracting complex numbers, it's important to handle the real and imaginary parts separately.
Real parts are treated just like normal numbers in arithmetic operations.
By identifying the real parts first, we make it easier to perform subtraction correctly.
This step ensures our results are accurate and logically organized.
imaginary parts
The imaginary part of a complex number involves the imaginary unit i. In \(9 + 5i\), the imaginary part is \(5i\).
In \(2 + 3i\), the imaginary part is \(3i\).
The imaginary unit i represents the square root of -1.
When performing operations on complex numbers, like addition or subtraction, treat the imaginary parts just like the real parts.
This means handling the coefficients of i separately from the real parts.
So for \5i - 3i\, the result is \2i\.
Always combine like terms to simplify expressions involving the imaginary unit.
subtracting complex numbers
Subtracting complex numbers involves both the real and imaginary parts.
First, identify and handle the real parts: \(9 - 2 = 7\).
Next, do the same for the imaginary parts: \(5i - 3i = 2i\).
Finally, combine these results to form the new complex number.
In our example, we combine 7 and \(2i\) to get \(7 + 2i\). Always ensure each part is subtracted correctly to avoid errors.
This method ensures we maintain the proper form of a complex number, written as \(a + bi\).
simplification
Simplification ensures that complex number expressions are as straightforward as possible.
After performing operations, combine like terms to write the result in standard form: \(a + bi\).
In our example, the final expression is \(7 + 2i\), which is already simplified.
Make sure the real part (a) and the imaginary part (b) are written clearly.
Simplification helps verify that the solution is both accurate and easy to understand.
It also allows for clearer communication of the result in both written and spoken forms.

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